T  F 


UC-NRLF 


B    3   131 


FIELD-BOOK   FOR 
RAILROAD   ENGINEERS 


CIRCULAR    AND    PARABOLIC   CURVES, 
TURNOUTS,    VERTICAL    CURVES,    LEVELLING, 

COMPUTING    EARTH-WORK, 
TRANSITION   CURVES   ON   NEW    LINES   AND 

APPLIED  TO    EXISTING   LINES, 

TOGETHER   WITH   TABLES   OF   RADII,    ORDINATES, 

LONG   CHORDS,    LOGARITHMS,    LOGARITHMIC 

AND   NATURAL   SINES,    TANGENTS,    ETC., 

AND    A   METRIC   CURVE   TABLE 


BY 

JOHN    B.    HENCK,   A.  M. 

LATE    PROFESSOR    OF    CIVIL    ENGINEERING    IN    THE    MASSACHUSETTS 
INSTITUTE    OF    TECHNOLOGY 


NEW   YORK    AND    LONDON 

D.  APPLETON   AND    COMPANY 

1912 


COPYRIGHT,  1854,  1881,  1896, 
BY  D.  APPLETON  AND  COMPANY 


Copyright,  1909,  by  John  B.  Henck,  Alice  C.  Henck,  and 
Edward  W.  Henck 


Printed  in  the  United  States  of  America 


PREFACE. 


IN  revising  this  work  for  the  second  time,  the  original 
purpose  of  making  the  volume  compact,  so  as  to  be  of  con- 
venient size  for  use  in  the  field,  has  been  adhered  to.  It  is 
designed  to  contain  such  formulae  and  tables  as  are  mat- 
ters of  constant  reference  in  the  field,  to  the  exclusion  of 
such  as  are  rarely  used.  Subjects  that,  though  important 
in  themselves,  require  large  space  for  satisfactory  treat- 
ment, or  are  best  learned,  once  for  all,  in  the  office  or  from 
competent  superiors  in  the  field,  are  also  excluded.  The 
size  of  the  volume  will  therefore  be  found  not  materially 
increased  by  the  changes  and  additions  now  made. 

Table  I.  has  been  enlarged.  The  first  column  contains 
the  degrees  of  curves  for  evary  two  minutes  up  to  10°,  for 
every  four  minutes  up  to  20°,  and  for  every  ten  minutes 
afterward.  The  deflection  angles  will  thus  be  always 
whole  minutes.  Ordinates  for  the  quarter  points,  both  for 
100  feet  chords  and  for  30  feet  rails,  are  new  features.  The 
column  of  chord  deflections  has  been  omitted,  being 
easily  supplied  by  doubling  the  tangent  deflections.  All 
the  data  required  in  laying  out  a  curve  are  found  on  one 
line.  Some  changes  have  been  made  in  the  other  tables, 
and,  in  connection  with  the  short  metric  curve  table,  a 
method  is  given  of  extending  it  by  means  of  Tables  I.,  II., 
III.,  and  IV.  The  length  of  the  arc  of  a  curve  is  seldom 
required,  since  a  curve  is  sufficiently  described  by  giving  the 
number  and  length  of  the  chords  and  the  deflection  angle 

iii 

387476 


IV  PREFACE. 

used.  When  the  length  of  the  arc  is  desired,  it  may  be 
found  by  the  method  given  in  §  13,  which  is  exact  for 
curves  laid  out  with  chords  of  any  length. 

Matters  formerly  in  an  Appendix  have  been  transferred 
to  their  proper  places  in  the  text.  Some  of  them  have 
been  more  fully  developed,  especially  those  relating  to 
turnouts  tangent  to  the  main  line. 

Transition  curves  have  been  more  fully  treated,  and  by 
methods  entirely  new.  These  curves  have  assumed  great 
importance  in  view  of  the  high  speed  of  modern  trains. 
The  shock  on  entering  and  leaving  a  curve,  and  the  dan- 
ger of  derailment,  may  be  greatly  reduced  by  a  transition 
curve,  if  carefully  located  and  laid  with  rails  that  have 
been  accurately  curved.  Both  these  essentials  are  secured 
by  the  methods  here  given.  Certain  portions  of  the  dis- 
cussion involve  the  calculus,  but  the  actual  laying  out  of 
the  curve  merely  requires  the  engineer  to  fix  upon  the 
length  of  curve  he  deems  best,  after  which  all  the  data  for 
locating  the  curve,  either  by  tangent  offsets  or  by  deflec- 
tion angles,  are  found  on  a  single  line  of  a  short  table. 
The  method  of  applying  a  transition  curve  to  an  existing 
track  is  equally  simple.  The  deflection  angle  of  the  exist- 
ing circular  curve  and  its  tangent  point  being  known,  and 
the  length  of  the  proposed  transition  curve  chosen,  a  single 
line  of  a  short  table  gives  the  data  for  locating  the  curve. 
In  this  table  the  ratio  of  the  two  radii  concerned  is  taken 
as  .9,  but  the  general  formulae  are  not  confined  to  any  par- 
ticular ratio.  It  will  be  seen  that  these  methods  do  not 
require  the  central  circular  curve  to  be  of  some  whole 
degree.  The  deflection  angle  D  of  the  central  curve  may 
have  any  value  we  please — a  manifest  advantage. 

For  curving  the  rails  accurately  the  ordinates  at  the 
centre  and  at  the  quarter  points  are  required.  These  are 
readily  found,  especially  when  the  curve  is  made  to  begin 
at  a  joint. 

The  chapter  on  the  common  parabola  is  retained,  be- 
cause, though  this  curve  has  met  with  but  little  acceptance 
on  railroads,  it  is  well  adapted  to  vertical  curves,  and  also 


PREFACE.  V 

affords  a  simple  means  of  laying  out  curves  on  common 
roads  and  pleasure  drives,  and  such  as  are  used  in  land- 
scape gardening. 

In  the  first  preface  to  this  work  (1854)  it  was  said: 
u  Among  the  processes  believed  to  be  original  may  be  speci- 
fied those  in  §§  41-48,  on  Compound  Curves,  in  Chapter 
II.,  011  Parabolic  Curves,  in  §§  106-109  (now  149-151)  on 
Vertical  Curves,  and  in  the  article  on  Excavation  and 
Embankment.  It  is  but  just  to  add  that  a  great  part  of 
what  is  said  on  Reversed  Curves,  Turnouts,  and  Crossings, 
and  most  of  the  Miscellaneous  Problems,  are  the  result  of 
original  investigations."  The  claims  here  made  have  been 
properly  recognized  by  some  authors,  while  others  have 
thought  it  sufficient  to  acknowledge  the  merits  of  the  pro- 
cesses involved  by  simply  adopting  them. 

J.  B.  H. 

MOKTECITO,  CAL.,  January,  1896. 


TABLE   OF   CONTENTS. 


CHAPTEK  I. 

CIRCULAR    CURVES. 

ARTICLE  I. — SIMPLE  CURVES. 

SECT.  PAGE 

2.  Definitions.     Propositions  relating  to  the  circle  ....  1 

4.  Angle  of  intersection  and  radius  given,  to  lind  the  tangent        .  3 

5.  Angle  of  intersection  and  tangent  given,  to  find  the  radius        .  3 

6.  Degree  of  a  curve 4 

7.  Deflection  angle  of  a  curve 4 

A.    Method  by  Deflection  Angles. 

9.  Kadius  given,  to  find  the  deflection  angle 5 

10.  Deflection  angle  given,  to  find  the  radius 5 

11.  Angle  of  intersection  and  tangent  given,  to  find  the  deflection 

angle    .        . 5 

12.  Angle  of  intersection  and  deflection   angle  given,  to  find  the 

tangent 6 

13.  Angle  of  intersection  and  deflection  angle  given,  to  find  the 

length  of  the  curve 6 

14.  Deflection  angle  given,  to  lay  out  a  curve 7 

16.  To  find  a  tangent  at  any  station 9 

B.     Method  by  Tangent  and  Chord  Deflections. 

17.  Definitions 9 

18.  Kadius  given,  to  find  the  tangent  deflection  and  chord  deflection  9 

19.  Deflection  angle  given,  to  find  the  chord  deflection    .        .        .10 

20.  To  find  a  tangent  at  any  station 11 

21.  Chord  deflection  given,  to  lay  out  a  curve    .        .        -        ,        .11 


Viii  TABLE   OF   CONTENTS. 

C.    Method  by  Offsets  from  Tangent. 

SECT.  PAGE 

23.  Deflection  angle  given,  to  find  points  on  the  curve  by  offsets 

from  the  tangent 13 

D.     Ordinates. 

24.  Definition 15 

25.  Deflection  angle  or  radius  given,  to  find  ordinates       .        .        .16 

26.  Approximate  value  for  middle  ordinate 18 

27.  Method  of  finding  intermediate  points  on  a  curve  approximately     18 

E.     Curving  Hails. 

29.  Deflection  angle  or  radius  given,  to  find  the  ordinate  for  curv- 

ing rails .19 

ARTICLE  II. — EEVERSED  AND  COMPOUND  CURVES. 

30.  Definitions 19 

31.  Kadii  or  deflection  angles  given,  to  lay  out  a  reversed,  or  com- 

pound curve 20 

A.     Reversed  Curves. 

32.  Reversing  point  when  the  tangents  are  parallel   .        .        .        .20 

33.  To  find  the  common  radius  when  the  tangents  are  parallel        .     21 

34.  One  radius  given,  to  find  the  other  when  the  tangents  are  par- 

allel         21 

35.  Chords  given,  to  find  the  radii  when  the  tangents  are  parallel    .  22 

36.  Radii  given,  to  find  the  chords  when  the  tangents  are  parallel  .  23 

37.  Common  radius  given,  to  run  the  curve  when  the  tangents  are 

not  parallel 23 

38.  One  radius  given,  to  find  the  other  when  the  tangents  are  not 

parallel         ...........     24 

39.  To  find  the  common  radius  when  the  tangents  are  not  parallel.     25 

40.  Second  method  of  finding  the  common  radius  when  the  tan- 

gents are  not  parallel .26 

B.     Compound  Curves. 

41.  Common  tangent  point  of  the  two  arcs 27 

42.  To  find  a  limit  in  one  direction  of  each  radius     .        .        .        .28 

44.  One  radius  given,  to  find  the  other 29 

45.  Second  method  of  finding  one  radius  when  the  other  is  given   .  31 

46.  To  find  the  two  radii 32 

47.  To  find  the  tangents  of  the  two  branches 34 

43.  Second  method  of  finding  the  tangents  of  the  two  branches       ,  35 


TABLE   OF   CONTENTS. 


ARTICLE  III. — TURNOUTS  AND  CROSSINGS. 
SECT.  PAGE 

49.  Three  cases  of  turnouts 36 

First  and  Second  Cases. 

50.  Definitions 37 

A.     Turnout  from  Straight  Main  Track. 

51.  Radius  given,  to  find  the  frog  angle  and  the  position  of  the  frog  37 

52.  Frog  angle  given,  to  find  the  radius  and  the  position  of  the  frog  38 

53.  To  find  mechanically  the  proper  position  of  a  given  frog    .        .  39 

54.  To  find  the  second  radius  of  a  turnout  reversing  opposite  the 

frog 40 

B.     Crossings  on  Straight  Lines. 

55.  Keferences  to  proper  problems .42 

56.  Kadii  given,  to  find  the  distance  between  switches      .        .        .42 

C.     Turnout  from  Curves. 

57.  Frog  angle  given,  to  find  the  radius  of  the  turnout  and  the  posi- 

tion of  the  frog 43 

58.  To  find  mechanically  the  proper  position  of  a  given  frog    .        .    47 

59.  Position  of  a  frog  given,  to  find  the  frog  angle     .        .        .        .47 

60.  Radius  of  turnout  given,  to  find  the  frog  angle  and  the  position 

of  the  frog 48 

62.  Turnout  to  reverse  and  become  parallel  to  the  main  track  .        .     51 

D.     Crossings  on  Curves. 

63.  Eeferences  to  proper  problems 52 

64.  Common  radius  given,  to  find  the  central  angles  and  chords      .     53 

Third  Case. 
Turnouts  Tangent  to  Main  Track. 

65.  Proper  length  of  switch-rail 53 

A.     Turnout  from  Straight  Lines. 

66.  Kadius  given,  to  find  the  frog  angle  and  the  position  of  the  frog     54 

67.  Frog  angle  given,  to  find  the  radius  and  the  position  of  the  frog     54 

68.  Locating  a  turnout  curve       ,        .        ,        t        ,        ,        t       f    55 


X  TABLE    OF    CONTENTS. 

B.     Crossings  on  Straight  Lines. 

SECT.  PAGE 

70.  References  to  proper  problems       .......     51 

C.     Turnout  from  Curves. 

71.  Frog  angle  given,  to  find  the  radius  of  the  turnout  and  the  po- 

sition of  the  frog 56 

72.  Eadius  of  the  turnout  given,  to  find  the  frog  angle  and  the  po- 

sition of  the  frog .        .        .        .        .        .        .        .        .        .59 

74.  Turnout  to  reverse  and  become  parallel  to  the  main  track  .        .     62 

75.  Position  of  a  frog  given,  to  find  the  frog  angle    .        .        .        .63 

D.     Crossings  on  Curves. 

76.  Eeferences  to  proper  problems 63 

E.     Double  Turnouts. 

77.  Those  turning  opposite  ways  and  those  turning  the  same  way  .     63 

78.  Finding  certain  chords,  frog  angles,  and  degrees  of  turnouts      .     65 

ARTICLE  IV. — MISCELLANEOUS  PROBLEMS. 

79.  To  find  the  radius  of  a  curve  to  pass  through  a  given  point       .     66 

80.  To  find  the  tangent  point  of  a  curve  to  pass  through  a  given 

point 67 

81.  To  find  the  distance  to  the  curve  from  any  point  on  the  tangent  67 

82.  Second  method  for  passing  a  curve  through  a  given  point .        .  67 

83.  To  find  the  proper  chord  for  any  angle  of  deflection  .        .        .  68 

84.  To  find  the  radius  when  the   distance  from   the   intersection 

point  to  the  curve  is  given 69 

85.  To  find  the  external,  that  is,  the  distance  from  the  intersection 

point  to  the  curve  when  the  radius  is  given     .        .        .        .70 

86.  To  find  the  tangent  point  of  a  curve  that  shall  pass  through  a 

given  point 70 

87.  To  find  the  radius  of  a  curve  without  measuring  angles      .        .  71 

88.  To  find  the  tangent  points  of  a  curve  without  measuring  angles  72 

89.  To  find  the  angle  of  intersection  and  the  tangent  points  when 

the  point  of  intersection  is  inaccessible 73 

90.  To  lay  out  a  curve  when  obstructions  occur         .         .        .        .76 

91.  To  change  the  tangent  point  of  a  curve,  so  that  it  may  pass 

through  a  given  point 77 

92.  To  change  the  radius  of  a  curve,  so  that  it  may  terminate  in  a 

tangent  parallel  to  its  present  tangent 78 

9§,  To  find  the  radius  of  a  curve  on  a/  track  already  laid  ,        ,        .79 


TABLE   OF   CONTENTS.  XI 


SECT- 

94.  To  draw  a  tangent  to  a  given  curve  from  a  given  point      •        .  80 

95.  To  flatten  the  extremities  of  a  sharp  curve   .....  80 

96.  To  locate  a  curve  without  setting  the  instrument  at  the  tangent 

point     ............  82 

97.  To  measure  the  distance  across  a  river          .....  84 

98.  To  change  a  tangent  point  so  that  the  tangent  may  pass  through 

a  given  point       ..........  86 

99.  To  connect  two  curves  by  a  common  tangent      .        .        .        .87 


CHAPTER  II. 

PARABOLIC    CURVES. 

ARTICLE  I. — LOCATING  PARABOLIC  CURVES. 

100.  Propositions  relating  to  the  parabola 89 

101.  To  lay  out  a  parabola  by  tangent  deflections        .        .        .        .90 

102.  To  lay  out  a  parabola  by  middle  ordinates 91 

103.  To  draw  a  tangent  to  a  parabola 92 

105.  To  lay  out  a  parabola  by  bisecting  tangents 93 

106.  To  lay  out  a  parabola  by  intersections 93 

107.  Example  illustrating  preceding  methods 94 

ARTICLE  II. — RADIUS  OF  CURVATURE. 

108.  Definition 95 

109.  To  find  the  radius  of  curvature  at  certain  stations       .        .        .96 

110.  Example  in  finding  radius  of  curvature 99 

111.  Simplification  when  the  tangents  are  equal 101 

112.  Length  of  parabolic  arcs 102 


CHAPTER  III. 

TRANSITION    CURVES. 

113.  Object  of  transition  curves 104 

ARTICLE  I. — THE  CUBIC  PARABOLA. 

114.  The  equation  of  the  cubic  parabola 104 

115.  Two  preliminary  problems  to  be  considered         ....  106 

116.  Angle  of  intersection  and  radius  of  central  curve  given,  to  find 

the  tangent 106 

117.  Angle  of  intersection  and  tangent  given,  to  find  the  radius  of 

the  central  curve        ,       t       ,,,,.,,  107 


Xil  TABLE    OF    CONTENTS. 

SECT.  PAGE 

118.  Length  of  the  abscissa  ^  of  the  transition  curve          .        .        .  108 

119.  Formulae  when  the  abscissa  xl  is  expressed  in  rail  lengths  of  30 

feet 108 

120.  Laying  out  the  transition  curve  by  offsets    .        .  .        .  110 

121.  Table  A.— Data  for  the  method  by  offsets    .        .  .        .110 

122.  Example  when  R  or  D  is  given Ill 

123.  Example  when  Misgiven Ill 

124.  Laying  out  the  transition  curve  by  deflection  angles  .        .        .112 

124.  Table  B.— Data  for  the  method  by  deflection  angles  .        .        .113 

125.  Example  of  the  method  by  deflection  angles        ....  113 

ARTICLE  II. — THE  CUBIC  PARABOLA  APPLIED  TO  AN  EXISTING  CIRCULAR 
TRACK. 

126.  Necessary  formulae  deduced   ........  113 

127.  Table  C. — Data  for  applying  the  cubic  parabola  to  an  existing 

track 115 

128.  Example  of  cubic  parabola  applied  to  an  existing  track      .        .  116 

129.  Length  of  transition  curve  in  terms  of  its  chords         .        .        .  116 

ARTICLE  III. — CURVING  THE  KAILS. 

131.  Ordinates  for  curving  the  rails  of  a  transition  curve    .        .        .  118 

ARTICLE  IV. — COMPOUND  TRANSITION  CURVE. 

132.  Coordinates  of  stations  on  a  compound  transition  curve.      .        .  119 

133.  Two  preliminary  problems  to  be  considered         ....  120 

134.  Angle  of  intersection  and  radius  of  central  curve  given,  to  find 

the  tangent 120 

135.  Example  when  angle  of  intersection  and  radius  of  central  curve 

are  given 122 

136.  Angle  of  intersection  and  tangent  given,  to  find  the  radius  of  the 

central  curve 122 

137.  Advantage  of  beginning  a  transition  curve  at  a  joint  .        .        .  123 


CHAPTER   IV. 

LEVELLING. 

ARTICLE  I. — HEIGHTS  AND  SLOPE  STAKES. 

138.  Definitions 124 

139,  To  find  the  difference  of  level  of  two  points         ,        t        f        .324 


TABLE    OF   CONTENTS. 

PAGE 

SECT< 

140.  Datum  plane ^ 

141.  To  find  the  heights  of  the  stations  on  a  line         .        .        •        •  1< 

142.  Sights  denominated  plus  and  minus 12^ 

143.  Form  of  field  notes ^ 

144.  To  set  slope  stakes 15 

ARTICLE  II.— CORRECTION  FOR  THE  EARTH'S  CURVATURE  AND  FOR 
KEFRACTION. 

-I  01 

145.  Earth's  curvature 

1  ^1 

146.  Refraction • 

147.  To  find  the  correction  for  curvature  and  refraction      .        .        .1* 

ARTICLE  III.— VERTICAL  CURVES. 

148.  Manner  of  designating  grades         .        .        •        v       •        •        •  ^ 

149.  To  find  the  grades  for  a  vertical  curve  at  whole  stations     .        .  le 

151.  To  find  the  grades  for  a  vertical  curve  at  sub-stations         .        .135 

ARTICLE  IV.— ELEVATION  OF  THE  OUTER  KAIL  ON  CURVES. 

152.  To  find  the  proper  elevation  of  the  outer  rail        .        .        •        •  1< 

153.  Coning  of  the  wheels      . 137 

ARTICLE  V.— EASING  GRADES  ON  CURVES. 

154.  Resistance  on  curves  and  grades  compared 138 

ARTICLE  VI. — EXPANSION  OF  RAILS. 

155.  Formula  for  the  proper  distance  between  rails      ....  139 


CHAPTER  V. 

EARTH-WORK. 

ARTICLE  I. — PRISMOIDAL  FORMULA. 

156.  Definition  of  a  prismoid 140 

157.  To  find  the  solidity  of  a  prismoid 140 

ARTICLE  II. — BORROW-PITS. 

158.  Manner  of  dividing  the  ground 141 

159.  To  find  the  solidity  of  a  vertical  prism  whose  horizontal  section 

is  a  triangle 142 


Xiv  TABLE   OF   CONTENTS. 

SECT.  PAGE 

160.  To  find  the  solidity  of  a  vertical  prism  whose  horizontal  section 

is  a  parallelogram 143 

161.  To  find  the  solidity  of  a  number  of  adjacent  prisms  having  the 

same  horizontal  section 144 

ARTICLE  III. — EXCAVATION  AND  EMBANKMENT. 
A.     Centre  Heights  alone  given. 

163.  To  find  the  solidity  of  one  section 145 

164.  To  find  the  solidity  of  any  number  of  successive  sections  .        .  146 

B.     Centre  and  Side  Heights  given. 

165.  Mode  of  dividing  the  ground 148 

166.  To  find  the  solidity  of  one  section 148 

167.  To  find  the  solidity  of  any  number  of  successive  sections  .        .  152 

169.  To  find  the  solidity  when  the  section  is  partly  in  excavation 

and  partly  in  embankment         .        .        .  .        .        .154 

170.  Beginning  and  end  of  an  excavation 156 

C.     Ground  very  Irregular. 

171.  To  find  the  solidity  when  the  ground  is  very  irregular       .        .  156 

172.  Usual  modes  of  calculating  excavation  examined         .        .        .  158 

D.     Correction  in  Excavation  on  Curves. 

173.  Nature  of  the  correction 159 

174.  To  find  the  correction  in  excavation  on  curves     .        .        .        .160 
176.  To  find  the  correction  when  the  section  is  partly  in  excavation ' 

and  partly  in  embankment 161 

178.  Note  on  the  computation  of  earthwork 163 


TABLES. 

I.    Eadii,  Ordinates,  Tangent  Deflections,  and  Ordinates  for 

Curving  Rails 165 

II.    Long  Chords 174 

III.  Tangents  and  Externals  of  a  One-degree  Curve    .        .        .  176 

IV.  Corrections  for  Table  III 183 

V.     Turnouts  Tangent  to  a  Straight  Main  Track.        .        .        .183 

VI.    Length  of  Circular  Arcs  in  Parts  of  Radius  ....  184 


TABLE   OF   CONTENTS.  XV 

NO.  PAGE 

VII.    Elevation  of  the  Outer  Kail  on  Curves 184 

VIII.  Correction  for  the  Earth's  Curvature  and  for  Refraction      .  185 

IX.     Rise  per  Mile  of  Various  Grades 186 

X.  Trigonometrical  and  Miscellaneous  Formula.        .        .        .188 

XI.    Heights  by  Aneroid  Barometer 194 

XII.    Heights  by  Aneroid  Barometer 201 

XIII.  Squares,  Cubes,  Square  Eoots,  Cube  Roots,  and  Reciprocals  203 

XIV.  Logarithms  of  Numbers  . » 221 

XV.  Logarithmic  Sines,  Cosines,  Tangents,  and  Cotangents        .  237 

XVI.  Natural  Sines  and  Cosines 286 

XVII.  Natural  Tangents  and  Cotangents 295 

XVIII.  Comparison  of  French  and  English  Weights  and  Measures.  308 

XIX.    Metric  Curve  Table 309 


EXPLANATION  OF  SIGNS. 


THE  sign  +  indicates  that  tfte  quantities  between  which  it  is 
placed  are  to  be  added  together. 

The  sign  —  indicates  that  the  quantity  before  which  it  is  placed 
is  to  be  subtracted. 

The  sign  x  indicates  that  the  quantities  between  which  it  is 
placed  are  to  be  multiplied  together. 

The  sign  •+-  or  :  indicates  that  the  first  of  two  quantities  be- 
tween which  it  is  placed  is  to  be  divided  by  the  second. 

The  sign  =  indicates  that  the  quantities  between  which  it  is 
placed  are  equal. 

The  sign  oo  indicates  that  the  difference  of  the  two  quantities 
between  which  it  is  placed  is  to  be  taken. 

The  sign  . *.  stands  for  the  word  "  hence  "  or  "  therefore." 

The  ratio  of  one  quantity  to  another  may  be  regarded  as  the 
quotient  of  the  first  divided  by  the  second.  Hence,  the  ratio  of 
a  to  b  is  expressed  by  a  :  b,  and  the  ratio  of  c  to  d  by  c  :  d.  A  pro- 
portion expresses  the  equality  of  two  ratios.  Hence,  a  proportion  is 
represented  by  placing  the  sign  =  between  two  ratios ;  as,  a :  b  =  c :  d. 


In  the  text  and  in  the  tables  the  foot  has  been  taken  as  the  unit 
of  measure  when  no  other  unit  is  specified. 


FIELD-BOOK. 


CHAPTER  I. 
CIRCULAR  CURVES. 

ARTICLE  I.— SIMPLE  CURVES. 

1.  THE  railroad  curves  here  considered  are  either  Circular  or 
Parabolic.     Circular  curves  are  divided  into  Simple,  Reversed,  and 
Compound  Curves.     We  begin  with  Simple  Curves. 

2.  Let  the  arc  A  D  E  F  B  (fig.  1)  represent  a  railroad  curve, 


Fig.  1 


2  CIRCULAR   CURVES. 

uniting  the  straight  Ime,^  G,A  'and  JB II.  The  length  of  such  a 
curve  is  measured  by  chords,  each  1QO  feet  long.*  Thus,  if  the 
chords  AD,  RE, ;.E*F.  and  F  B  jvre  each  100  feet  in  length,  the 
whole  curve  is  said  to  be  400  feet  long*.  The  straight  lines  Cr  A  and 
B  H are  always  tangent  to  the  curve  at  its  extremities,  which  are 
called  tangent  points.  If  G  A  and  B  H  are  produced,  until  they 
meet  in  C,  A  C  and  B  C  are  called  the  tangents  of  the  curve.  If 
A  C  is  produced  beyond  G  to  K,  the  angle  KGB,  formed  by  one 
tangent  with  the  other  produced,  is  called  the  angle  of  intersec- 
tion, and  shows  the  change  of  direction  in  passing  from  one  tan- 
gent to  the  other. 

The  following  propositions  relating  to  the  circle  are  derived 
from  Geometry : 

I.  A  tangent  to  a  circle  is  perpendicular  to  the  radius  drawn 
through  the  tangent  point.     Thus,  A  G  is  perpendicular  to  A  0, 
and  B  C  to  B  0. 

II.  Two  tangents  drawn  to  a  circle  from  any  point  are  equal, 
and  if  a  chord  be  drawn  between  the  two  tangent  points,  the 
angles  between  this  chord  and  the  tangents  are  equal.     Thus 
A  C  =  B  <7,  and  the  angle  B  A  C  =  A  B  C. 

III.  An  acute  angle  between  a  tangent  and  a  chord  is  equal 
to  half  the  central  angle  subtended  by  the  same  chord.     Thus, 
CA  B  =  $AOB. 

IV.  An  acute  angle  subtended  by  a  chord,  and  having  its  vertex 
in  the  circumference  of  a  circle,  is  equal  to  half  the  central  angle 
subtended  by  the  same  chord.     Thus,  DAE  =  %DOE. 

V.  Equal  chords  subtend  equal  angles  at  the  centre  of  a  circle, 
and  also  at  the  circumference,  if  the  angles  are  inscribed  in  similar 
segments.     Thus,  A  0  D  =  D  0  E,  and  D  A  E  =  E  A  F. 

VI.  The  angle  of  intersection  of  two  tangents'  is  equal  to  the 
central  angle  subtended  by  the  chord  which  unites  the  tangent 
points.     Thus,  K  G  B  =  A  0  B. 

3.  In  order  to  unite  two  straight  lines,  as  Cr  A  and  B  H,  by  a 
curve,  the  angle  of  intersection  is  measured,  and  then  a  radius  for 
the  curve  may  be  assumed,  and  the  tangents  calculated,  or  the 

*  Some  engineers  prefer  a  chain  50  feet  in  length,  and  measure  the  length 
of  a  curve  by  chords  of  50  instead  of  100  feet.  The  chord  of  100  feet  has 
been  adopted  throughout  this  article  ;  but  the  formulae  deduced  may  be 
very  readily  modified  to  suit  chords  of  any  length.  See  also  §  13. 


SIMPLE    CURVES. 


tangents  may  be  assumed  of  a  certain  length,  and  the  radius  cal- 
culated. 

4.  Problem.      Given  the  angle  of  intersection  K  C  B  =  1 
(fig.  1)  and  the  radius  AO  —  R^to  find  the  tangent  A  C  =  T. 


Fig.  1. 


Solution.    Draw  C  0.    Then  in  the  right  triangle  A  0  C  we 

AC 
have  (Tab.  X.  3)  —  =  tan.  A  0  C,  or,  since  A  0  C  =  1 1  (§  2,  VI.) 

T 

-  =  tan.  i  7; 


Example.    Given  /=  22°  52',  and  R  =  3000,  to  find  T.    Here 


R  =  3000 
i  /=  11°  26' 

^=606.72 


3.477121 
tan.  9.305869 

2.7829UO 


CIRCULAR   CURVES. 


5.  Problem.      Given  the  angle  of  intersection 

(fig.  1)  and  the  tangent  A  C  =  T,  to  find  the  radius  A  0  =  R. 

Solution.    In  the  right  triangle  A  0  C  we  have  (Tab.  X.  6) 


.-.R=T  cot.  i  /. 
Example.    Given  /=  31°  16'  and  T=  950,  to  find  R.    Here 

T-950  2.977724 

i  I  =15°  38'  cot  0.553102 

R  =  3394.89  3.530826 

6.  The  degree  of  a  curve  is  determined  by  the  angle  subtended 
at  its  centre  by  a  chord  of  100  feet.     Thus,  if  A  OD  =  6°  (fig.  1), 
ADEF  B  is  a  6°  curve. 

7.  The  deflection  angle  of  a  curve  is  the  acute  angle  formed  at 
any  point  between  a  tangent  and  a  chord  of  100  feet.     The  deflec- 
tion angle  is,  therefore(§  2,  III.),  half  the  degree  of  the  curve.  Thus, 
GAD  or  CBFis  the  deflection  angle  of  the  curve  ADEFB, 
and  is  half  A  0  D  or  half  FOB. 

Remark.  The  mode  of  designating  curves  by  their  degree,  given 
above,  is  objected  t^  by  some,  because  when  curves  are  laid  out  by 
chords  shorter  than  100  feet,  as  is  usual  on  sharp  curves,  the  degree 
of  the  curve  is  slightly  increased,  though  its  designation  remains 
the  same.  If  the  arc  of  100  feet  is  substituted  for  the  chord  of  100 
feet  in  the  definition,  this  difficulty  vanishes  ;  but  so  many  greater 
difficulties  are  introduced  that  the  general  adoption  of  this  method 
is  not  probable.  Moreover,  when  American  engineers  use  the  met- 
ric system,  as  possibly  they  are  now  doing  on  Mexican  roads,  both 
these  methods  are  inapplicable.  We  might  designate  a  curve  by  the 
length  of  its  radius,  for  this  fixes  the  curve,  however  laid  out,  and 
any  units  of  length  may  be  used  ;  but  when  the  deflection  angle  D  is 
even,  R  is  generally  fractional,  which  makes  it  inconvenient  for  ex- 
act definition.  The  length  of  the  radius  is  also  an  indirect  desig- 
nation, when  curves  are  laid  out  by  deflection  angles.  If  the  curve 
were  designated  by  its  deflection  angle  for  a  certain  length  of  chord 
any  length  of  chord  and  any  units  of  length  might  be  used,  and  th' 
curve  be  still  definitely  described.  Thus  we  might  say  :  "  Curve  tx 
the  right,  deflection  angle  for  chords  of  50  feet,  2°  10',"  or,  "  Curve 
to  the  left,  deflection  angle  for  chords  of  20  metres,  1°  35'." 


METHOD  BY  DEFLECTION  ANGLES.  5 

A.  Method  by  Deflection  Angles. 

8.  The  usual  method  of  laying  out  a  curve  on  the  ground  is  by 
means  of  deflection  angles. 

9.  Problem.     Given  the  radius  AO  =  R  (fig.  1),  to  find  the 
deflection  angle  C  B  F  =  D. 

Solution.    Draw  OL  perpendicular  to  B  F.    Then  the  angle 
BOL  =  ^BOF=D^  and  B  L  =  $  B  F  =  50.    But  in  the  right 

71     T 

triangle  0  B  L  we  have  (Tab.  X.  1)  sin.  BOL=  — -  ; 

B  0 

'     D-— 

~~  R' 

Example.    Given  R  =  5729.65,  to  find  D.    Here 

50  1.698970 

R  =  5729.65  3.758128 


D  =  30'  sin.  7.940842 

Hence  a  curve  of  this  radius  is  a  1°  curve,  and  its  deflection 
angle  is  30'. 

10.  Problem.     Given  the  deflection  angle  CBF=  D  (fig.  1), 
to  find  the  radius  AO  =  R. 

Solution.    By  the  preceding  section  we  have  sin.  D  =  —  ,  whence 

R 

R  sin.  D  =  50  ; 


By  this  formula  the  radii  in  Table  I.  are  calculated. 

Example.    Given  D  —  1°,  to  find  R.    Here 

50  1.698970 

D=l°  sin.  8.241855 


R  =  2864.93  3.457115 

11.  Problem.  Given  the  angle  of  intersection  KCB  —  1 
(fig.  1),  and  the  tangent  AC  =  T,  to  find  the  deflection  angle 
CAD  =  D. 

50 

Solution.     From  §  9  we  have  sin.  D—  —  ,  and  from  §  5, 

H 


5  CIRCULAR   CURVES. 

R  =.  Tcot.  J  /.    Substituting  this  value  of  R  in  the  first  equa- 
_       5Q        . 

50  tan.  1 1 

.  • .  sm.  D  =  — . 


Example.    Given  J=  21°  and  T-  424.8,  to  find  D.    Here 

50  1.698970 

^  1  =  10°  30'  tan.  9.267967 


0.966937 
T-  424.8  2.628185 


D  =  1°  15'  sin.  8.338752 

12.  Problem.  Given  the  angle  of  intersection  KGB  —  1 
(fig.  1),  and  the  deflection  angle  CAD=D,  to  find  the  tangent 
AC=  T. 

Solution.     From   the   preceding   section   we   have   sin.    D  =f 

Hence,  T  sin.  D  =  50  tan.  *  J; 


sin.D 
Example.    Given  /=  28°  and  Z)  =  1°,  to  find  T.     Here 

°=  714.31. 


13.  Problem.  Given  the  angle  of  intersection  KCB  =  1 
(fig.  1),  and  the  deflection  angle  CAD  —  D,  to  find  the  length  of 
the  curve. 

Solution.  By  §  2  the  length  of  a  curve  is  measured  by  chords 
of  100  feet  applied  around  the  curve.  Now  the  first  chord  A  D 
makes  with  the  tangent  A  C  an  angle  G  A  D  =  Z>,  and  each  suc- 
ceeding chord  D  E,  E  F,  &c.  subtends  at  A  an  additional  angle 
DAE,EAF,  &c.,  each  equal  to  D  ;  since  each  of  these  angles 
(§  2,  IV.)  is  half  of  a  central  angle  subtended  by  a  chord  o±  100 
feet.  The  angle  CAB  =  \AOB  =  \I\$,  therefore,  made  up  of 
as  many  times  D,  as  there  are  chords  around  the  curve.  Then  if 
n  represents  the  number  of  chords,  we  have  n  D  =  •£/; 


If  D  is  not  contained  an  even  number  of  times  in  £  J,  the  quo- 
tient above  will  still  give  the  length  of  the  curve.     Thus,  in 


METHOD  BY  DEFLECTION  ANGLES.  7 

figure  2,  suppose  D  is  contained  4|  times  in  £  /.  This  shows  that 
there  will  be  four  whole  chords  and  f  of  a  chord  around  the  curve 
from  A  to  B.  The  angle  GAB,  the  fraction  of  Z>,  is  called  a 
sub-deflection  angle,  and  G  B,  the  fraction  of  a  chord,  is  called 
a  sub-chord* 

The  length  of  the  curve  thus  found  is  not  the  actual  length  of 
the  arc,  but  the  length  required  in  locating  a  curve.  If  the  actual 
length  of  the  arc  is  required,  it  may  be  found  by  means  of  Table  VI. 

Example.     Given/—  16°  52'  and  D  =  1°  20',  to  find  the  length  of 

the  curve.    Here  n  =^=r-  =  -^-577,  =  -^  TT  =  6.325,  that  is,  the  curve 
JLf        1    &0         oU 

is  632.5  feet  long. 

To  find  the  arc  itself  in  this  example,  we  take  from  Table  VI. 
the  length  to  radius  1  of  an  arc  of  16°  52',  since  the  central  angle 
of  the  whole  curve  is  equal  to  1  (§  2,  VI.),  and  multiply  this  length 
by  the  radius  of  the  curve. 

Arc  10°  =  .1745329 
"  6°  =  .1047198 
"  50'  =.0145444 
"  2'  =  .0005818 


"    16°  52'  =  .2943789 

The  radius  of  the  curve  is  found  from  Table  I.  to  be  2148.79,  and 
this  multiplied  by  .2943789  gives  632.558  feet  for  the  length  of 
the  arc. 

14.  Problem.     Given  the,  deflection  angle  D,  to  lay  out  a 

curve  from  a  given  tangent  point. 

Solution.  Let  A  (fig.  2)  be  the  given  tangent  point  in  the 
tangent  H  C.  Set  the  instrument  at  A,  and  lay  off  the  given  de- 
flection angle  D  from  A  C.  This  will  give  the  direction  A  D,  and 
100  feet  being  measured  from  A  in  this  direction,  the  point  D  will 
be  determined.  Lay  off  in  succession  the  additional  angles  DAE, 
EAF,  &c.,  each  equal  to  D,  and  make  D  E,  E  F,  &c.,  each  100 
feet,  and  the  points  E,  F,  &c.,  will  be  determined.  The  points 

*  This  method  of  finding  the  length  of  a  sub-chord  is  not  mathematically 
accurate  ;  for,  by  geometry,  angles  inscribed  in  a  circle  are  proportional  to 
the  arcs  on  which  they  stand  ;  whereas  this  method  supposes  them  to  be 
proportional  to  the  chords  of  these  arcs.  In  railroad  curves,  the  error 
arising  from  this  supposition  is  too  small  to  be  regarded. 


CIRCULAR   CURVES. 


Z>,  E,  F,  &c.,  thus  determined,  are  points  on  the  required  curve 
(§  7,  and  §  2,  III.,  IV.),  and  are  called  stations. 

If  there  is  a  sub-chord  at  the  end,  as  O  B,  the  sub-deflection 
angle  GAB  must  be  the  same  part  of  D  that  G  B  is  of  a  whole 


Fig.  2. 


chord  (§  13).  If  there  is  a  sub-chord  at  the  beginning,  the  first 
stake  on  the  curve  will  be  at  the  end  of  the  sub-chord,  and  the 
sub-deflection  angle  will  be  the  same  part  of  D  that  the  sub-chord 
is  of  a  whole  chord. 

In  laying  out  a  curve  there  is  an  obvious  advantage  in  having 
the  several  deflection  angles  whole  minutes.  When  the  deflection 
angle  is  assumed,  whole  minutes  would  naturally  be  chosen.  But 
when  D  is  found  from  /  and  I7  by  §  11,  it  generally  happens  that 
D  does  not  come  out  even  minutes.  In  such  cases,  unless  it  is 
necessary  that  the  curve  should  commence  exactly  at  the  assumed 
tangent  point,  it  is  better  to  take  D  to  the  nearest  minute,  and 
calculate  T  for  /  and  this  new  value  of  D  by  §  12.  If,  however, 
there  is  a  sub-chord  at  the  beginning  of  the  curve,  the  sub-deflec- 
tion angle  will  generally  contain  seconds,  although  D  contains 
none.  In  this  case,  set  the  vernier  back  the  amount  of  the  sub- 
deflection  angle,  so  that,  when  this  angle  is  turned  off,  the  instru- 
ment will  read  zero.  All  the  subsequent  angles  will  then  be  whole 
minutes. 


METHOD  BY  DEFLECTION  ANGLES.  9 

15.  It  is  often  impossible  to  lay  out  the  whole  of  a  curve,  with- 
out removing  the  instrument  from  its  first  position,  either  on  ac- 
count of  the  great  length  of  the  curve,  or  because  some  obstruction 
to  the  sight  may  be  met  with.     In  this  case,  after  determining  as 
many  stations  as  possible,  and  removing  the  instrument  to  the 
last  of  these  stations,  we  ought  to  be  able  to  find  the  tangent  to 
the  curve  at  this  station ;  for  then  the  curve  could  be  continued 
by  deflections  from  the  new  tangent  in  precisely  the  same  way  as 
it  was  begun  from  the  first  tangent. 

16.  Problem.     After  running  a  curve  a  certain  number  of 
stations,  to  find  a  tangent  to  the  curve  at  the  last  station. 

Solution.  Suppose  that  the  curve  (fig.  2)  has  been  run  three 
stations  to  F,  and  that  F  L  is  the  tangent  required.  Produce 
A  F  to  K,  and  we  have  the  angle  KF  L  =  A  F  C.  But  (§  2,  II.) 
AFC -F  AC.  Therefore  K  F  L  =  F  A  C.  Now  J'M  GT  is  the 
sum  of  all  the  deflection  angles  laid  off  from  the  tangent  at  A, 
that  is,  in  this  case,  F  A  C  =  3  Z>,  and  the  tangent  F  L  is,  there- 
fore, obtained  by  laying  off  from  A  ^produced  an  angle  K  F  L 
equal  to  the  total  deflection  from  the  preceding  tangent. 

If  the  curve  is  afterwards  continued  beyond  F,  as,  for  instance, 
to  B,  a  tangent  B  N  at  B  is  obtained  by  laying  off  from  F  B  pro- 
duced an  angle  MBN  =  LBF=LFB,  the  total  deflection 
from  the  preceding  tangent  F  L. 

B.    Method  by  Tangent  and  Chord  Deflections. 

17.  Let  A  B  CD  (fig.  3)  be  a  curve  between  the  two  tangents 
E  A  and  D  L,  having  the  chords  A  B,  B  C,  and  CD  of  the  same 
length.     Produce  the  tangent  E  A,  and  from  B  draw  B  G  per- 
pendicular to  A  G.     Produce  also  the  chords  A  B  and  B  C,  and 
make  the  produced  parts  B II  and  C  K  of  the  same  length  as  the 
chords.     Draw  C  H  and  D  K.    B  G  is  called  the  tangent  deflec- 
tion,  and  C  H  or  D  K  the  chord  deflection. 

18.  Problem.     Given  the  radius  AO  =  R  (fig.  3\  to  find  the 
tangent  deflection  B  G,  and  the  chord  deflection  C  H. 

Solution.  The  triangle  C  B  H  is  similar  to  BOG',  for  the 
angle  BOC  =  180°  -  (0  B  C  +  B  C  0\  or,  since  BCO  =  ABO, 
BOC=  180°  -  (0  B  C  +  A  B  0)  =  CB  H,  and,  as  both  the  tri- 
angles are  isosceles,  the  remaining  angles  are  equal.  The  ho- 


10 


CIRCULAR   CURVES. 


mologous  sides  are,  therefore,  proportional,  that  is,  B  0  :  B  C  = 
B  C  :  C  ff,  or,  representing  the  chord  by  c  and  the  chord  deflection 
by  d,  R  :  c  =  c  :  d  ; 


To  find  the  tangent  deflection,  draw  B  M  to  the  middle  of  C  H, 
bisecting  the  angle  C  B  H,  and  making  BMC  a  right  angle. 
Then  the  right  triangles  BMC  and  A  G  B  are  equal  ;  f  or  B  C  — 


Fig.  3. 


AB,  and  the  angle  C  B  M  =  J  OB  H  =  %  B  0  C  =  i  A  0  B  = 
£  .1  #  (§  2,  III.).  Therefore  B  G  =  C  M  =  %  C  H  =  %  d,  that  is, 
the  tangent  deflection  is  half  the  chord  deflection. 

19.   Problem.     Given  the  deflection  angle  D  of  a  curve,  to 
find  the  chord  deflection  d.  2 

Solution.    By  the  preceding  section  we  have  d  =  -=,  and  by 

50 

8  10,  R  =  - — =:.     Substituting  this  value  of  R  in  the  first  equa- 
sin.  D 

tion,  we  find 

c2  sin.  D 

~m-> 

This  formula  gives  the  chord  deflection  for  a  chord  c,  of  any  length, 
though  D  is  the  deflection  angle  for  a  chord  of  100  feet  (§  7).  When 
c  =  100,  the  formula  becomes  d  =  200  sin.  Z>,  or  for  the  tangent  de- 


METHOD  BY  TANGENT  AND  CHORD  DEFLECTIONS.     H 

flection  \  d  =  100  sin.  D.    By  this  formula  the  tangent  deflections 
in  Table  I.  may  be  easily  obtained  from  the  table  of  natural  sines. 
The  length  of  the  curve  may  be  found  by  first  finding  D  (§  9  or 
§  11),  and  then  proceeding  as  in  §  13. 

20.  Problem.     To  draw  a  tangent  to  the  curve  at  any  station, 

as  B  (fig.  3). 

Solution.  Bisect  the  chord  deflection  H  C  of  the  next  station 
in  M.  A  line  drawn  through  B  and  M  will  be  the  tangent  re- 
quired; for  it  has  been  proved  (§  18)  that  the  angle  C  B  M  is  in 
this  case  equal  to  i  B  0  (7,  and  B  M  is  consequently  (§  2,  III.)  a 
tangent  at  B. 

If  B  is  at  the  end  of  the  curve,  the  tangent  at  B  may  be  found 
without  first  laying  off  H  C.  Thus,  if  a  chain  equal  to  the  chord, 
is  extended  to  H  on  A  B  produced,  the  point  H  marked,  and  the 
chain  then  swung  round,  keeping  the  end  at  B  fixed,  until  H  M  = 
£  d,  B  M  will  be  the  direction  of  the  required  tangent.* 

21.  Problem.     Given  the  chord  deflection  d,  to  lay  out  a 

curve  from  a  given  tangent  point. 

Solution.  Let  A  (fig.  3)  be  the  given  tangent  point,  and  sup- 
pose d  has  been  calculated  for  a  chord  of  100  feet.  Stretch  a  chain 
of  100  feet  from  A  to  G  on  the  tangent  E  A  produced,  and  mark 
the  point  G.  Swing  the  chain  round  towards  A  B,  keeping  the  end 
at  A  fixed,  until  B  G  is  equal  to  the  tangent  deflection  i  d,  and  B 
will  be  the  first  station  on  the  curve.  Stretch  the  chain  from  B  to 
H  on  A  B  produced,  and  having  marked  this  point,  swing  the 
chain  round,  until  H  C  is  equal  to  the  chord  deflection  d.  C  is  the 
second  station  on  the  curve.  Continue  to  lay  off  the  chord  deflec- 
tion from  the  preceding  chord  produced,  until  the  curve  is  finished. 

Should  the  curve  begin  or  end  with  a  sub-chord,  denote,  as  be- 
fore, the  whole  chord  by  c,  the  sub-chord  by  c',  the  tangent  deflec- 
tion for  c  by  £  d,  and  that  for  c'  by  ±  d1.  Then  (§18)  i  d  =  £= 

aJ» 

C 

and  i  d'  =  ^.     Therefore  £  d  :  -J-  d'  =  c* :  c'2, 
&J& 

(c'  \2 
)• 

*  The  distance  B  M  is  not  exactly  equal  to  the  chord,  but  the  error  aris- 
ing from  taking  it  equal  is  too  small  to  be  regarded  in  any  curves  but  those 
of  very  small  radius.  If  necessary,  the  true  length  of  B  M  may  be  calcu- 
lated ;  f  or  B  M  -  V ITTT2  -  H  Jkf2. 


12 


CIRCULAR   CURVES. 


If  the  curve  begins  with  a  sub-chord,  produce  the  tangenfc  a 
distance  c',  and  from  its  extremity  lay  off  a  distance  *  d'  for  a 
point  on  the  curve.  But  as  we  need  a  whole  chord  in  order  to 
produce  it  for  continuing  the  curve,  measure  back  on  the  tangent 
a  distance  c  —  c'  =  c"  and  lay  off  the  deflection  proper  to  c",  but 
in  an  opposite  direction  to  •£  d'.  This  will  give  a  point  on  the 
curve  supposed  to  be  run  back  to  the  preceding  whole  station. 
The  line  joining  these  two  points  on  the  curve  will  now  be  a  whole 
chord,  and  can  be  produced  in  the  usual  way.  If  the  curve  ends 
in  a  sub-chord,  as  D  F  (fig.  3),  find  the  tangent  DL  (§  20),  and 
lay  off  from  it  the  proper  tangent  deflection  LF  for  the  sub- 
chord,  found  as  above. 


Fig.  3. 


Example.  Given  the  intersection  angle  /  between  two  tangents 
equal  to  16°  30',  and  R  —  1250,  to  find  T,  d,  and  the  length  of  the 
curve  in  stations.  Here 

(§4)    T=  R  tan.  *  7  =  1250  tan.  8M5'  =  181.24; 


(§  9)   sin.  D  =  ™  =  ^  =  .04  =  nat.  sin.  2°  17*' ; 


;'  -    495'   -S60 

77  —  19ry  r-/  —  rf-bl). 


METHOD   BY    OFFSETS   FROM   TANGENT.  13 

These  results  show,  that  the  tangent  point  A  (fig.  3)  on  the  first 
tangent  is  181.24  feet  from  the  point  of  intersection, — that  the 
tangent  deflection  G  B  =  i  d  =  4  feet,— that  the  chord  deflection 
HC  or  K  D  =  8  feet,— and  that  the  curve  is  360  feet  long.  The 
three  whole  stations  B,  C,  and  D  having  been  found,  and  the  tan- 
gent D  L  drawn,  the  tangent  deflection  for  the  sub-chord  of  60 

feet  will  be,  as  shown  above,  $d'  =  4  ( — j  =  4  x  .62  =  4  x  .36  = 

1.44.  L  F  =  1.44  feet  being  laid  off  from  D  L,  the  point  F  will, 
if  the  work  is  correct,  fall  upon  the  second  tangent  point.  A  tan- 
gent at  F  may  be  found  (§  20)  by  producing  D  F  to  P,  making 
FP  =  D  F  =  60  feet,  and  laying  off  PN  =  1.44  feet.  F  N  will 
be  the  direction  of  the  required  tangent,  which  should,  of  course, 
coincide  with  the  given  tangent. 

Curves  may  be  laid  out  with  accuracy  by  tangent  and  chord  de- 
flections, if  an  instrument  is  used  in  producing  the  lines.  But  if 
an  instrument  is  not  at  hand,  and  accuracy  is  not  important,  the 
lines  may  be  produced  by  the  eye  alone.  On  sharp  curves,  such 
as  sometimes  occur  on  street  railroads,  where  the  chords  may  not 
exceed  10  feet,  a  fine  cord  may  be  used  for  producing  the  lines. 
The  radius  of  a  curve  to  unite  two  given  straight  lines  may  also 
be  found  without  an  instrument  by  §  87,  or,  having  assumed  a  ra- 
dius, the  tangent  points  may  be  found  by  §  88. 


C.  Method  ly  Offsets  from  Tangent. 

22.  By  this  method  points  on  a  curve  such  as  C  (fig.  3a)  are  de- 
termined by  measuring  from  the  tangent  point  certain  distances 
along  the  tangent,  such  as  A  B,  and  offsets  at  right  angles  to  the 
tangent,  such  as  B  C. 

23.  Problem.     Given  D,  the  deflection  angle  of  a  curve  for  a 
chord  c,  to  find  A  B  =  a  (fig.  3d)  and  B  C  =  b  for  a  point  C  on 
the  curve,  distant  from  the  tangent  point  a  certain  number  of 
stations,  whole  or  fractional,  denoted  ~by  the  letter  n. 

Solution.  The  angle  B  A  C  =  n  D,  and  the  central  angle 
A  0  C  =  2  n  D.  Draw  C  D  parallel  to  the  tangent.  Then,  in  the 
triangle  CD  0,  we  have 

a  =  CD  =  C  0  sin.  DOC-R  sin.  2  n  D. 


14 


CIRCULAR   CURVES. 


Substituting  for  R  its  value   .         , 

|  c  sin.  2  n  D 




sin.  D 
To  find  b,  we  have 

b  =  £C  =  AO-DO  =  R-R  cos.  2  n  D,  or  (Tab.  X.,  23) 
b  =  R  —  R  (1-2  sin.2  rc  D}  =  2R  sin2  n  D. 

Substituting  for  R  its  value   .         , 

c  sin.2  71 D 


In  computing  these  values  for  successive  points,  the  logarithms 
(  remain  constant,  which  facilitates  the  work. 


:  and  of 


OI  — ; f\  MIIJI^A    v/j.   — ^.    . 

sm.  D  sin.  D 

The  position  of  the  stakes  is  best  fixed  by  measuring  the  successive 
chords,  instead  of  depending  on  the  right  angle  at  B. 

If  the  offsets  from  the  original  tangent  become  inconveniently 
long,  a  new  tangent  is  readily  found.     Thus  a  tangent  T  C  at  C 

is  determined  by  measuring  from 
Fig.  3a.  A  a  distance  A  T  =  R  tan.  n  D  = 

A  T  R  i c  tan.  n  D     m      ,      . , 

- — : — ^=r — .    T  C  should,  of  course, 
sm.  D 

prove  equal  to  A  T. 

Since  n  may  be  a  fraction  or  a 
mixed  number,  as  well  as  a  whole 
number,  n  c  may  represent  any  sub- 
chord,  such  as  would  generally  oc- 
cur at  the  beginning  of  a  curve. 
The  points  on  the  curve  determined 
by  the  formulas  for  a  and.  b  will 
therefore  be  the  regular  stations 
continued  from  the  straight  line. 

In    laying   out   a   whole    curve 

AEB  (fig.  3&)  by  this  method  a  tangent  D  G  at  the  middle  point 
of  the  curve  is  found  by  computing  the  equal  distances  A  Z), 
D  E,  E  G,  and  G  B  by  the  formula  AD-DE-EG-GB- 
R  tan.  J  /.  As  a  check,  the  distance  C  E  may  be  found  from  the 
triangle  C  E  D.  For  C  E  —  D  E  tan.  £  /.  Substituting  for  D  E 
its  value  R  tan.  i  /,  we  have  C  E  =  R  tan.  |  /tan.  £  /. 
The  station  of  the  tangent  point  A  being  known,  and  the  length 


OBDINATES. 


15 


of  the  curve  having  been  found  (§  13),  the  stations  of  E  and  B  are 
readily  found.  Then,  by  the  process  just  explained,  find  the  off- 
sets from  the  tangent  A  D  to  the  regular  stations  on,  say,  one 


Fig.  36. 


quarter  of  the  curve.  By  the  same  process,  beginning  at  the 
known  station  at  E,  find  offsets  to  the  regular  stations  on  the 
curve.  In  like  manner,  offsets  from  the  tangents  E  G  and  B  (r 
will  complete  the  curve,  the  regular  stations  being  kept  through- 
out. Curves  may  be  laid  out  with  great  accuracy  by  this  method. 

D.  Ordinates. 

24.  The  preceding  methods  of  laying  out  curves  determine 
points  100  feet  distant  from  each  other.  These  points  are  usually 
sufficient  for  grading  a  road ;  but  when  the  track  is  laid,  it  is  de- 
sirable to  have  intermediate  points  on  the  curve  accurately  deter- 
"hined.  For  this  purpose  the  chord  of  100  feet  is  divided  into  a 


16 


CIRCULAR   CURVES. 


certain  number  of  equal  parts,  and  the  perpendicular  distances 
from  the  points  of  division  to  the  curve  are  calculated.  These 
distances  are  called  ordinates. 


25.  Problem.  Given  the  deflection  angle  D  or  the  radius  R 
of  a  curve,  to  find  the  ordinates  for  any  chord. 

Solution.  I.  To  find  the  middle  ordinate.  Let  AE  B  (fig.  4) 
be  a  portion  of  a  curve,  subtended  by  a  chord  A  B,  which  may  be 


G 


denoted  by  c.     Draw  the  middle  ordinate  E  Z>,  and  denote  it  by 
m.    Produce  E  D  to  the  centre  F,  and  join  A  F  and  A  E.    Then 


(Tab.  X.  3)  =p^  =  tan.  E  A  D,  or  E  D  =  A  D  tan.  E  A  D.     But, 

A.  JLJ 

since  the  angle  E  A  D  is  measured  by  half  the  arc  B  E,  or  by  half 
the  equal  arc  A  E,  we  have  E  AD  =  %AFE.  Therefore  E  D  = 
ADtsin.lt  A  FE,  or 


When  c  =  100,  A  F  E  =  D  (§  7),  and  m  =  50  tan.  $  D,  whence 
m  may  be  obtained  from  the  table  of  natural  tangents,  by  divid- 
ing tan.  \  D  by  2,  and  removing  the  decimal  point  two  places  to 
the  right. 

The  value  of  m  may  be  obtained  in  another  form  thus:  In  the 


ORDINATES.  17 

triangle  A  D  F  we  have  D  F  =  \/A  F*  -  A  D*  - 
Then  m  —  E  F  —  DF=R  —  D  F,  or 

m  =  R  — 


II.  To  find  any  other  ordinate,  as  R  jV,  at  a  distance  D  N  =  b 
.from  the  centre  of  the  chord.  Produce  R  N  until  it  meets  the 
diameter  parallel  to  A  B  in  G,  and  join  R  F.  Then  R  G  = 


-  t>\  and  RN=RG-NG=RG- 
D  F.  Substituting  the  value  of  R  G  and  that  of  D  F  found 
above,  we  have 


The  other  ordinates  may  also  be  found  from  the  middle  ordi- 
nate by  the  following  shorter,  but  not  strictly  exact  method.  It 
is  founded  on  the  supposition,  that,  if  the  half-chord  B  D  be 
divided  into  any  number  of  equal  parts,  the  ordinates  at  these 
points  will  divide  the  arc  E  B  into  the  same  number  of  equal 
parts,  and  upon  the  further  supposition,  that  the  tangents  of  small 
angles  are  proportional  to  the  angles  themselves.  These  suppo- 
sitions give  rise  to  no  material  error  in  finding  the  ordinates  of 
railroad  curves  for  chords  not  exceeding  100  feet.  Making,  for 
example,  four  divisions  of  the  chord  on  each  side  of  the  centre, 
and  joining  A  R,  A  £,  and  A  T,  we  have  the  angle  JRAN  = 
IE  AD,  since  R  B  is  considered  equal  to  J  E  B.  But  E  A  D  = 
\AFE.  Therefore,  R  A  N  =  f  A  F  E.  In  the  same  way  we 
should  find  SA  0  =  J  A  F  E,  and  TAP=\A  F  E.  We  have 
then  for  the  ordinates,  RN=A  N  tan.  R  A  N  =  |  c  tan.  f  A  F  E, 


I  c  tan.  ^AFE.  But,  by  the  second  supposition,  tan.  f  A  F  E  = 
f  tan.  i  A  FE,  tan.  J  A  FE  =  |  tan.  $A  F  E,  and  tan.  |  A  FE  = 
±ta,u.$AFE.  Substituting  these  values,  and  recollecting  that 
i  c  tan.  |  A  F  E  —  m,  we  have 

=       x  ictan.  \ 


S  0  =  -j  x  i  c  tan.  £  A  FE  =  -^  m, 

7  7 

jT-P  =  ^-5  x  ^  c  tan.  ^  ^4.  F  E  =  77;  w. 
lo  lo 

In  general,  if  the  number  of  divisions  of  the  chord  on  each  side 
3 


}g  CIRCULAR    CURVES. 

of  the  centre  is  represented  by  n,  we  should  find  for  the  respect- 

(n  +  1)  (n  —  1)  m 
ive    ordinates,   beginning   nearest   the   centre,  -       —  -^  —     —  , 

(n  +  2)  (n  -  2)  m  (n  +  3)(n-3)m  ^ 

M*  tf 

These  values  of  the  ordinates  are  precisely  what  we  should  ob- 

tain if  we  regarded  A  E  B  as  the  arc  of  a  parabola  ;  for  in  this 

vcase,  as  we  shall  see  later,  the  offsets  from  a  tangent  at  E  to  R,  S, 

and   T  would  be  ^  m,  ^  m,  and  ^  m.     Subtracting  these  dis- 
lo        lo  lo 

tances  from  m,  we  should  get  the  results  given  above. 

Example.  Find  the  ordinates  of  an  8°  curve  to  a  chord  of  100 
feet.  Here  m  =  50  tan.  2°  =  1.746,  RN  =  ~.m  =  1.637,  S  0  = 
|  w  =  1.310,  and  TP  =  ^  m  =  0.764. 

26.  An  approximate  value  of  m  also  may  be  obtained  from  the 
formula  m  =  R  —  \/  R*  —  i  c§.  This  is  done  by  adding  to  the 

c4 
quantity  under  the  radical  the  very  small  fraction  .    ^  ,  making 

c2 
it  a  perfect  square,  the  root  of  which  will  be  R  —  ^-~.    We  have, 

then,  m  -  R  -    R  -  ~    ; 


27.  From  this  value  of  m  we  see  that  the  middle  ordinates  of 
any  two  chords  in  the  same  curve  are  to  each  other  nearly  as  the 
squares  of  the  chords.     If,  then,  A  E  (fig.  4)  be  considered  equal 
to  |  A  B,  its  middle  ordinate  C  H  =  J  E  D.     Intermediate  points 
on  a  curve  may,  therefore,  be  very  readily  obtained,  and  generally 
with  sufficient  accuracy,  in  the  following  manner  :  Stretch  a  cord 
from  A  to  J5,  and  by  means  of  the  middle  ordinate  determine  the 
point  E.     Then  stretch  the  cord  from  A  to  E,  and  lay  off  the 
middle  ordinate  C  H  =  %  E  D,  thus  determining  the  point  C,  and 
so  continue  to  lay  off  from  the  successive  half-chords  one-fourth 
the  preceding  ordinate,  until  a  sufficient  number  of  points  is  ob- 
tained. 

E.  Curving  Rails. 

28.  The  rails  of  a  curve  are  usually  curved  before  they  are  laid. 
To  do  this  properly,  it  is  necessary  to  know  the  middle  ordinate 


REVERSED  AND  COMPOUND  CURVES.  19 

of  the  curve  for  a  chord  of  the  length  of  a  rail,  and  the  ordinates 
at  the  quarter  points. 

29.  Problem.  Given  the  radius  or  deflection  angle  of  a 
curve,  to  find  the  middle  ordinate  for  curving  a  rail  of  given 
length. 

Solution.  Denote  the  length  of  the  rail  by  Z,  and  we  have  (§  25) 
the  exact  formula  m  =  R  —  \/jR*  —  £  /2,  and  (§  26)  the  approxi- 
mate formula 


This  formula  is  always  near  enough  for  chords  of  the  length  of  a 

50 
rail.    If  we  substitute  for  R  its  value  (§  10)  R  =  ^  —  ^  ,  we  have, 


Example.  In  a  1°  curve  find  the  ordinate  for  a  rail  30  feet  in 
length. 

For  a  rail  30  feet  in  length  £  Z2  =  225,  and,  consequently,  m  = 
fl.25  sin.  D.  This  gives  for  a  1°  curve,  m  =  .02. 

The  corresponding  ordinate  for  a  curve  of  any  other  degree  may 
be  found  approximately  by  multiplying  the  ordinate  for  a  1°  curve 
by  the  number  expressing  the  degree  of  the  curve.  The  ordinates 
from  the  chord  at  the  quarter  points  are  (§  25)  each  f  m.  In  Table 
I.  are  given  the  values  of  m  and  £  m  for  a  rail  of  30  feet.  From 
these  ordinates  the  ordinates  for  a  rail  of  any  other  length  are  ob- 
tained by  simply  multiplying  by  the  square  of  the  ratio  of  its 
length  to  30.  Thus  for  a  rail  of  27  feet  this  ratio  is  .9,  the  square 
of  which  is  .81,  and  the  ordinates  for,  say,  a  4°  curve,  are  .079  x 
.81  =  .064  and  .059  x  .81  =  .048. 

ARTICLE  II.  —  REVERSED  AND  COMPOUND  CURVES. 

30.  Two  curves  often  succeed  each  other  having  a  common  tan- 
gent at  the  point  of  junction.  If  the  curves  lie  on  opposite  sides 
of  the  common  tangent,  they  form  a  reversed  curve,  and  their 
radii  may  be  the  same  or  different.  If  they  lie  on  the  same  side 
of  the  common  tangent,  they  have  different  radii,  and  form  a  com- 
pound curve.  Thus  ABC  (fig.  5)  is  a  reversed  curve,  and  ABB 
a  compound  curve. 


20  CIRCULAR   CURVES. 

81.  Problem.  To  lay  out  a  reversed  or  a  compound  curve, 
when  the  radii  or  deflection  angles  and  the  tangent  points  are 
known. 

Solution.  Lay  out  the  first  portion  of  the  curve  from  A  to  B 
(fig.  5),  by  one  of  the  usual  methods.  Find  B  F,  the  tangent  to 


A  B  at  the  point  B  (§  16  or  §  20).  Then  B  F  will  be  the  tangent 
also  of  the  second  portion  B  C  of  a  reversed,  or  B  D  of  a  com- 
pound curve,  and  from  this  tangent  either  of  these  portions  may 
be  laid  off  in  the  usual  manner. 

A.  Reversed  Curves. 

32.  Theorem.  The  reversing  point  of  a  reversed  curve  be- 
tween parallel  tangents  is  in  the  line  joining  the  tangent  points. 

Demonstration.  Let  A  C  B  (fig.  6)  be  a  reversed  curve,  uniting 
the  parallel  tangents  H  A  and  B  K,  having  its  radii  equal  or  un- 
equal, and  reversing  at  C.  If  now  the  chords  A  C  and  C  B  are 
drawn,  we  have  to  prove  that  these  chords  are  in  the  same  straight 
line.  The  radii  E  C  and  G  F,  being  perpendicular  to  the  common 
tangent  at  C  (§  2,  I.),  are  in  the  same  straight  line,  and  the  radii 
A  E  and  B  F,  being  perpendicular  to  the  parallel  tangents  H  A 
and  B  K,  are  parallel.  Therefore,  the  angle  A  E  C  =  C  F  B,  and, 
consequently,  EGA,  the  half  supplement  of  A  E  C\  is  equal  to 
F  C  B,  the  half  supplement  of  CFB;  but  these  angles  cannot 
be  equal,  unless  A  C  and  C  B  are  in  the  same  straight  line. 


REVERSED   CURVES.  21 

33.  Problem.  Given  the  perpendicular  distance  between 
two  parallel  tangents  B  D  =  b  (fig.  6),  and  the  distance  between 
the  two  tangent  points  A  B  =  a,  to  determine  the  reversing  point 
C  and  the  common  radius  EC=CF=R  of  a  reversed  curve 
uniting  the  tangents  H  A  and  B  K. 


Solution.  Let  A  C  B  be  the  required  curve.  Since  the  radii 
are  equal,  and  the  angle  A  E  C  =  B  F  (7,  the  triangles  AEG 
and  B  F  C  are  equal,  and  AC=CB  =  $a.  The  reversing  point 
C  is,  therefore,  the  middle  point  of  A  B. 

To  find  R,  draw  E  G  perpendicular  to  A  C.  Then  the  right 
triangles  AEG  and  BAD  are  similar,  since  (§  2,  III.)  the  angle 
BAD  =  \AEC  =  AEG.  Therefore  AE  :  AG  =  AB:BD, 
or  R\±a  —  a:b; 


Corollary.    If  R  and  b  are  given,  to  find  a,  the  equation  R  = 

a* 

j-j-  gives  a2  =  4  R  b  ; 

4  0 

HT  .  • .  a  =  2\/R~b. 

Examples.     Given  b  =  12,  and  a  =  200,  to  determine  R.    Here 
_    200*    _  10000  _ 

— —   1 TH  —  — To —  —  OOO'j. 

Given  R  =  675,  and  b  =  12,  to  find  a.    Here  a  —  2^/675  x  12  = 
2^8100  =  2  x  90  =  180. 

34.   Problem.     Given  the  perpendicular  distance  between  two 
parallel  tangents  B  D  =  b  (fig.  7),  the  distance  between  the  two 


CIBCULAR   CURVES. 


tangent  points  A  B  =  a.  and  the  first  radius  E  C  =  R  of  a  re- 
versed curve  uniting  the  tangents  H  A  and  B  £,  to  find  the  chords 
A  C  =  a'  and  C B  =  a",  and  the  second  radius  OF—  R'. 


Solution.  Draw  the  perpendiculars  EG  and  F L.  Then  the 
right  triangles  A  B  D  and  E  A  G  are  similar,  since  the  angle 
BAD  =  \AEG-AEG.  Therefore  AB\BD^EA\ AG, 
or  a  :  b  =  R  :  %  a' ; 

2Rb 


Since  a'  and  a"  are  (§  32)  parts  of  a,  we  have 
ft^~  a"  =  a  —  a'. 

To  find  R'   the  similar  triangles    ABD    and    FB  L  give 
AB-.BD  =  FB\  BL,  or  a:  b  =  R' :  |  a"; 

a  a" 


Example.    Given  b  =  8,  a  =  160,  and  R  =  900,  to  find  a',  a", 

n   v   QOO    v   ft 

-  =  90,  a"  =  160  -  90  =  70,  and 


and  R.    Here  a'  = 


35.  Corollary  1.    If  6,  a',  and  a"  are  given,  to  find  a, . 
and  R',  we  have  (§  34) 

,        ,,  ^      aa'          „,      a  a" 

a  =  a  +  a   -,        ±t  =  ^-r  ;        H 


REVERSED    CURVES. 


Example.    Given  6  =  8,  a'  =  90,  and  a"  =  70,  to  find 
160  x  70 


a,  R, 


1fiO  v  QO 

Here  a  =  90  +  70  =  160,  R  =  "         ™  =  900,  and  . 


2x8 


2x8 


.  =  700. 


36.  Corollary  2.  If  ^,  -R',  and  6  are  given,  to  find  a,  a', 
and  a",  we  have  (§  35),  R  +  R'  =  —a  ofta<—  =  -°  o  ^  ^  —  jfV 
Therefore  a2  =  2  6  (JK  +  J2') ; 

Jg^~  .-.»=• 

Having  found  a,  we  have  (§  34) 


Example.    Given  /jJ  =  900,  R'  =  700,  and  b  =  8,  to  find 
and    a".      Here    a  =  \/2  x  8(900  +  700)'=  V16  x  1600 

2  x  900  x  8  2  x  700  x  8 

a'  =  --       —  =  90,anda    =  --      -      -70. 


a,  a', 
160, 


37.   Problem.      Given  the  angle  AKB-=.K,  which  shows 
the  change  of  direction  of  two  tangents  HA  and  B  K  (fig.  8\  to 


-N 


B 

Fig.  8. 


unite  these  tangents  by  a  reversed  curve  of  given  common  radius 
R,  startina  from  a  given  tangent  point  A. 

Solution.  With  the  given  radius  run  the  curve  to  the  point  D, 
where  the  tangent  D  jV  becomes  parallel  to  B  K.  The  point  D  is 
found  thus.  Since  the  angle  N  B  K,  which  is  double  the  angle 


24  CIRCULAR   CURVES. 

H  A  D  (§  2,  II.),  is  to  be  made  equal  to  A  KB  =  K,  lay  off  from 
H  A  the  angle  HA  D  =  \K.  Measure  in  the  direction  thus  found 
the  chord  AD  =  2R  sin.  |  K.  This  will  be  shown  (§  83)  to  be 
the  length  of  the  chord  for  a  deflection  angle  i  K.  Having  found 
the  point  D,  measure  the  perpendicular  distance  D  M  —  b  between 
the  parallel  tangents. 

The  distance  BD  =  2DC=a  may  then  be  obtained  from  the 
formula  (§  33,  Cor.) 


The  second  tangent  point  B  and  the  reversing  point  C  are  now 
determined.    The  direction  of  D  B  or  the  angle  B  D  N  may  also 

be  obtained ;  for  sin.  B  D  N  =  sin.  D  B  M  =     - ,  or 


sin.BDN=-. 
a 


38.  Problem.  Given  the  line  A  B  —  a  (fig.  9\  which  joins 
the  fixed  tangent  points  A  and  B,  the  angles  ffAB  =  A  and 
A  B  L  =  B,  and  the  first  radius  A  E  —  R,  to  find  the  second 
radius  B  F  =  R'  of  a  reversed  curve  to  unite  the  tangents  H'  A 
and  B  K. 


Fig.  9. 


First  Solution.  With  the  given  radius  run  the  curve  to  the 
point  D,  where  the  tangent  D  N  becomes  parallel  to  B  K.  The 
point  D  is  found  thus.  Since  the  angle  U  G  N,  which  is  double 


KEVERSED   CURVES.  25 

H  A  D  (§  2,  II.),  is  equal  to  Av*B,  lay  off  from  HA  the  angle 
H  A  D  —  \(Acr>  B],  and  measure  in  this  direction  the  chord  AD  = 
2  R  sin.  \(Av*B)  (§  83). 

Setting  the  instrument  at  /),  run  the  curve  to  the  reversing  point 
C  in  the  line  from  D  to  B  (§  32),  and  measure  D  C  and  C  B. 
Then  the  similar  triangles  DEC  and  B  F  C  give  D  C  :  D  E  = 
CB-.BF,  or  DC:R=  CB:R'\ 

W  .'.R'=^x  R. 

Second  Solution.  By  this  method  the  second  radius  may  be 
found  by  calculation  alone.  The  figure  being  drawn  as  above,  we 
have,  in  the  triangle  A  B  D,  A  B  =  a,  A  D  =  2  R  sin.  %(A  —  B}^ 
and  the  included  angle  D  A  B  =  HA  B  —  H  A  D  =  A  -  % 
(A  —  B}  =  4  (A  +  B}.  Find  in  this  triangle  (Tab.  X.  14  and 
12)  B  D  and  the  angle  A  B  D.  Find  also  the  angle  D  B  L  =  B 
+  ABD. 

Then  the  chord  CB  =  2  R'  sin.  4  BFC  =  2  R'  sin.  D  B  L, 
and  the  chord  DC  =  2R  sin.  |  D  E  C  =  2  R  sin.  Z>  B  L  (§  83). 
But  CB  =  BD-DC;  whence  2  .#'  sin.  D  B  L  =  B  D  - 
2  R  sin. 


When  the  point  D  falls  on  the  other  side  of  J.,  that  is,  when 
the  angle  B  is  greater  than  A,  the  solution  is  the  same,  except 
that  the  angle  D  A  B  is  then  180°  -  i  (A  +  5),  and  the  angle 
DBL  =  B-  ABD. 

39.  Problem.  Given  the  length  of  the  common  tangent 
D  6r  =  a,  and  the  angles  of  intersection  I  and  I'  (fig.  10\  to  deter- 
mine the  common  radius  C  E  =  C  F  =  R  of  a  reversed  curve  to 
unite  the  tangents  HA  and  B  L. 

Solution.  By  §  4  we  have  D  C  =  R  tan.  }  7,  and  C  G  = 
R  tan.  £  /',  whence  R  (tan.  4  /  +  tan.  J  /')  =  D  C  +  (7  tf  =  a,  or 


This  formula  may  be  adapted  to  calculation  by  logarithms ;  for  we 

have  (Tab.  X.  35)  tan.  4  /  +  tan.  4  I  =    sm- ^J  +  ^)        Substi- 

cos.  4 /cos.  4 /' 
tilting  this  value,  we  get 

„  _  a  cos.  4  /cos.  4  /' 
sin.  4  (/  +  /') 


26  CIRCULAR    CURVES. 

The  tangent  points  A  and  B  are  obtained  by  measuring  from 

D  a  distance  A  D  =  R  tan.  •£  /,  and  from  G  a  distance  B  O  = 
R  tan.  |  /'. 

Example.    Given  a  =  600, 1  =  12°,  and  /'  =  8°,  to  find  R.    Here 

a  =  600  2.778151 

£7=6°  cos.  9.997614 

iJT  =4°  cos.  9.998941 


2.774706 
i  (/+/')  =  10°  sin.  9.239670 

R  =  3427.96          3.535036 

40.  Problem.  Given  the  line  A  B  =  a  (fig.  JO),  which  joins 
the  fixed  tangent  points  A  and  B,  the  angle  D  A  B  =  A,  and  the 
angle  A  B  Gr  =  B,  to  find  the  common  radius  E  C '  =  C  JP  —  R  of 
a  reversed  curve  to  unite  the  tangents  HA  and  B  L. 


Fig.  10. 


Solution.  Find  first  the  auxiliary  angle  A  KE  =  B  K  F, 
w/iich  may  be  denoted  by  K.  For  this  purpose  the  triangle  A  E  K 
gives  AE\EK  =  sin.  K :  sin.  E  A  K.  Therefore  E  JTsin.  K  = 
A  E  sin.  E  A  K  =  R  cos.  A,  since  E  A  K  =  90°  —  A.  In  like 
manner,  the  triangle  BF If  gives  FKsiu.  K=  HI1  sin.  FBK= 
R  cos.  B.  Adding  these  equations,  we  have  (E  K  +  F K)  sin.  K  = 
R  (cos.  A  +  cos.  B\  or,  since  EK+FK=2R,  2Rsin.K  = 


COMPOUND   CURVES.  27 

R  (cos.  A  +  cos.  B).  Therefore,  sin.  K  =  i  (cos.  A  +  cos.  B).  For 
calculation  by  logarithms,  this  becomes  (Tab.  X.  28) 

jgf"  sin.  K  =  cos.  |  (A  +  B)  cos.  i  ( J.  —  B}. 

Having  found  K,  we  have  the  angle  AEK=E  =  180°  —  K— 
E AK=  180°  —  K—  (90°  —  -4)  =  90°  +  A  —  K,  and  the  angle 
B  FK=  F=  180°  -  K-  FBK=  180°  -  K-  (90°  -  B}  =  90°  + 
B  —  K.  Moreover,  the  triangle  A  E  K  gives  A  E  :  A  K  = 
sin.  K :  sin.  E,  or  72  sin.  E  =  A  TTsin.  7T,  and  the  triangle  B  F  K 
gives  B  F  \BK—  sin.  7T :  sin.  F,  or  72  sin.  F  =  B  K  sin.  7T. 
Adding  these  equations,  we  have  R  (sin.  7?  +  sin.  F)  =  (A  K  + 
B  K)  sin.  K=a  sin.  7T.  Substituting  for  sin.  E  +  sin.  T^7  its 
value  2  sin.  |  (E  +  jP)  cos.  *  (E  -  F)  (Tab.  X.  26),  we  have 
2  72  sin.  ^(E  +  F)  cos.  ^  (E  —  F)  —  a  sin.  7T.  Therefore  72  = 

_  *  T_s — -= ^-.     Finally,  substituting    for  E   its 

sin.  i(E+  F)  cos.  1  (E  -  F) 

value  90°  +  A  —  K,  and  for  F  its  value  90°  4-  B  —  K,  we  get 
%(E  +  F)  =  90°  —  [K—  %(A  +  B)],  and  \  (E  —  F}  =  |  (A  —  B)\ 
whence 

__  i  a  sin.  K 

~  cos.  [JST-  i  (A  +  £)]  cos.  $(A  —  B)' 

Example.    Given  a  =  1500,  A  =  18°,  and  B  =  6°,  to  find  72. 

Here  *  (A  +  B)  — 12°  cos.  9.990404 

i  U  -  5)  =  6°  cos.  9.997614 

7T  =  76°  36'  10' '  sin.  9.988018 

i  a  =  750  2^875061 


2.863079 

-  ±  (A  +  B}  =  64°  36'  10"  cos.  9.632347 
i  (A  -  B)  =  6°       cos.  9.997614 

9.629961 


^  =  1710.48  3.233118 

B.  Compound  Curves. 

41.  Theorem.  If  one  branch  of  a  compound  curve  be  pro- 
duced, until  the  tangent  at  its  extremity  is  parallel  to  the  tangent 
at  the  extremity  of  the  second  branch,  the  common  tangent  point  of 
the  two  arcs  is  in  the  straight  line  produced,  which  passes  through 
the  tangent  points  of  these  parallel  tangents. 


28 


CIRCULAR    CURVES. 


Demonstration.  Let  ACB  (fig.  11)  be  a  compound  curve, 
uniting  the  tangents  H  A  and  B  K.  The  radii  C  E  and  C  F,  be- 
ing perpendicular  to  the  common  tangent  at  C  (§  2,  I.),  are  in  the 


Fig.  11. 


same  straight  line.  Continue  the  curve  A  G  to  D,  where  its  tan- 
gent 0  D  becomes  parallel  to  B  K,  and  consequently  the  radius 
D  E  parallel  to  B  F.  Then  if  the  chords  CD  and  G  B  be  drawn, 
we  have  the  angle  CED=CFB\  whence  E  C  D,  the  half- 
supplement  of  C  E  D,  is  equal  to  F  C  B,  the  half-supplement  of 
C  FB.  But  E  C  D  cannot  be  equal  to  F  C  B,  unless  CD  coin- 
cides with  CB.  Therefore  the  line  BD  produced  passes  through 
the  common  tangent  point  C. 

42.  Problem.  To  find  a  limit  in  one  direction  of  each  radius 
of  a  compound  curve. 

Solution.  Let  A  I  and  B  I  (fig.  11)  be  the  tangents  of  the 
curve.  Through  the  intersection  point  /,  draw  I M  bisecting  the 


COMPOUND   CURVES.  29 

angle  A  1 B.  Draw  A  L  and  B  M  perpendicular  respectively  to 
A  /and  B  I,  meeting  I M  in  L  and  M.  Then  the  radius  of  the 
branch  commencing  on  the  shorter  tangent  A  1  must  be  less  than 
A  *L,  and  the  radius  of  the  branch  commencing  on  the  longer  tan- 
gent B I  must  be  greater  than  B  M.  For  suppose  the  shorter 
radius  to  be  made  equal  to  A  L,  and  make  I N '  =  A  7,  and  join 
L N.  Then  the  equal  triangles  AIL  and  NIL  give  A  L  = 
L  N;  so  that  the  curve,  if  continued,  will  pass  through  N,  where 
its  tangent  will  coincide  with  IN.  Then  (§  41)  the  common  tan- 
gent point  would  be  the  intersection  of  the  straight  line  through 
B  and  N  with  the  first  curve ;  but  in  this  case  there  can  be  no 
intersection,  and  therefore  no  common  tangent  point.  Suppose 
next,  that  this  radius  is  greater  than  A  L,  and  continue  the  curve, 
until  its  tangent  becomes  parallel  to  B  I.  In  this  case  the  ex- 
tremity of  the  curve  will  fall  outside  the  tangent  B I  in  the  line 
A  N  produced,  and  a  straight  line  through  B  and  this  extremity 
will  again  fail  to  intersect  the  curve  already  drawn.  As  no  com- 
mon tangent  point  can  be  found  when  this  radius  is  taken  equal 
to  A  L  or  greater  than  A  L,  no  compound  curve  is  possible.  This 
radius  must,  therefore,  be  less  than  A  L.  In  a  similar  manner  it 
might  be  shown,  that  the  radius  of  the  other  branch  of  the  curve 
must  be  greater  than  B  M.  If  we  suppose  the  tangents  A  I  and 
B I  and  the  intersection  angle  I  to  be  known,  we  have  (§  5)  A  L  — 
A  I  cot.  -$•  /,  and  B  M  =  B I  cot.  |  /.  These  values  are,  therefore, 
the  limits  of  the  radii  in  one  direction. 

43.  If  nothing  were  given  but  the  position  of  the  tangents  and 
the  tangent  points,  it  is  evident  that  an  indefinite  number  of  dif- 
ferent compound  curves  might  connect  the  tangent  points;  for 
the  shorter  radius  might  be  taken  of  any  length  less  than  the 
limit  found  above,  and  a  corresponding  value  for  the  greater  could 
be  found.     Some  other  condition  must,  therefore,  be  introduced, 
as  is  done  in  the  following  problems. 

44.  Problem,     Given  the  line  A  B  =  a  (fig.  11),  which  joins 
the  fixed  tangent  points  A  and  B,  the  angle  B  A  1=  A,  the  angle 
A  B 1=  B,  and  the  first  radius  A  E  =  R,  to  find  the  second  ra- 
dius B F  =  R'  of  a  compound  curve  to  unite  the  tangents  HA 
and  B  K. 

Solution.    Suppose  the  first  curve  to  be  run  with  the  given 
radius  from  A  to  Z),  where  its  tangent  D  0  becomes  parallel  to 


30 


CIRCULAR   CURVES. 


B  /,  and  the  angle  I A  D  =  |  ( A  +  B).  Then  (§  41)  the  common 
tangent  point  C  is  in  the  line  B  D  produced,  and  the  chord  C  B  = 
C  D  +  B  D.  Now  in  the  triangle  A  B  D  we  have  A  B  =^  a, 


Fig.  11. 


>  =  2  R  sin.  $(A  +  B)  (§  83),  and  the  included  angle  D  A  B  = 
7  J.  jB  —  / A  D  =  A  —  £  (A  +  .5)  =  \(A  —  B).  Find  in  this  tri- 
angle (Tab.  X.  14  and  12)  the  angle  A  B  D  and  the  side  B  D. 
Find  also  the  angle  CBI=B  —  ABD. 

Then  (§  83)  the  chord  C  B  =  2R'  sin.  C  B  I,  and  the  chord 
C  D  —  2  R  sin.  CDO  =  2R  sin.  C  B  I.  Substituting  these  values 
otCB  and  CD  in  the  equation  found  above,  C  B  =  CD  +  B  D, 
we  have  2  R'  sin.  CBI=2Rsin.  C  B I  + 

BD 


"2  sin. 

When  the  angle  B  is  greater  than  A,  that  is,  when  the  greater 
radius  is  given,  the  solution  is  the  same,  except  that  the  angle 
>  —  A),  and  C  B  I  is  found  by  subtracting  the  sup- 


COMPOUND   CURVES.  31 

plement  of  ABD  from  B.    We  shall  also  find  CB  =  CD  — 

7?  T) 
B  Z>,  and  consequently  R  '  =  R  — 


. 

&  sin. 

If  more  convenient,  the  point  D  may  be  determined  in  the  field, 
by  laying  off  the  angle  I  A  D  =  %  (A  +  B),  and  measuring  the 
distance  A  D  =  2  R  sin.  ±(A  +  B).  BD  and  C  B  I  may,  then  be 
measured,  instead  of  being  calculated  as  above. 

Example.  Given  a  =  950,  A  =.  8°,  B  =  7°,  and  ft  =  3000,  to 
find  R'.  Here  A  D  -  2  x  3000  sin.  i  (8°  +  7°)  =  783.16,  and 
D  A  B  =  i  (8°  —  7°)  =  30'.  Then  to  find  A  B  D  we  have 

A  B  -  A  D  =  166.84  2.222300 

i  (A  D  B  +  A  B  D)  =  89°  45'  tan.  2.360180 

4.582480 
A  B  +  A  D  =  1733.16  3.238839 

i  (A  D  B  -  A  B  D)  -  87°  24'  17"    tan.  1.343641 

.  •  .  A  B  D  =  2°  20'  43" 
Next,  to  find  B  D, 

AD  =  783.16  2.893849 

D  A  B  =  30'  sin.  7.940842 


0.834691 
A  B  D  =  2°  20'  43"      sin.  8.611948 


B  D  =  167.01  2.222743 

B-ABD=CBI=4°W  17"      sin.  8.909292 


2  (R  '  -  R)  =  2058.03  3.313451 

R'  -.#=:  1029.01 
R1  =  3000  +  1029.01  =  4029.01 

To  find  the  central  angle  of  each  branch,  we  have  CFB  = 
2  CBI=  9°  18'  34",  which  is  the  central  angle  of  the  second 
branch;  and  A  EC  =  AED  -  CED  =  A  +  B  -  2CBI  = 
5°  41'  26",  which  is  the  central  angle  of  the  first  branch. 

45.  Problem.  Given  (fig.  11)  the  tangents  AI=  T,BI  = 
T',  the  angle  of  intersection  —  /,  and  the  first  radius  A  E  =  R, 
to  find  the  second  radius  B  F  =  R' . 

Solution.  Suppose  the  first  curve  to  be  run  with  the  given  ra- 
dius from  A  to  D,  where  its  tangent  D  0  becomes  parallel  to  B  I. 
Through  D  draw  D  P  parallel  to  A  J,  and  we  have  /  P  =  D  0  = 


32  CIRCULAR   CURVES. 

AO  =  Rtfm.iI  (§4).  Then  in  the  triangle  DPB  we  have 
DP=  10=.  AI-AO=  T-  12  tan.*/,  BP=B1-1P= 
T1  -  R  tan.  |  /,  and  the  included  angle  DPB=AIB=  180°  - 
/.  Find  in  this  triangle  the  angle  C  B  I,  and  the  side  B  D.  The 
remainder  of  the  solution  is  the  same  as  in  g  44-  The  determina- 
tion of  the  point  D  in  the  field  is  also  the  same,  the  angle  IAD 
being  here  =  |  /.  When  B  is  greater  than  A,  that  is,  when  the 
greater  radius  is  given,  the  solution  is  the  same,  except  that  DP  = 
/  —  T,  and  BP=  R  tan.  j-  /—  T'. 


Example.  Given  T=  447.32,  T'  =  510.84,  7-  15°,  and  R  = 
3000,  to  find  ^'.  Here  ^  tan.  |  /  =  3000  tan.  71°  =  394.96,  D  P  = 
447.32  -  394.96  =  52.36,  J5  P  =  510.84  -  394.96  =  115.88,  and 
DPD  =  180°  -  15°  =  165°.  Then  (Tab.  X.  14  and  12) 

BP-DP-  63.52  1.802910 

i  (B  D  P  +  PB  D)  =  7°  30'  tan.  9.119429 

0.922339 
B  P  +  D  P  =  168.24  2.225929 


-  PB  D)  =  2°  50'  44"  tan.  8.696410 
.-.PBD=  CB  7=4°  39'  16" 
Next,  to  find  5  Z>, 

DP  =52.36  1.71900C 

DPB-  165°  sin.  9.412996 

1.13199C 

=  4°  39'  16"  sin.  8.90926^ 


B  D  =  167.005  2.222730 

The  tangents  in  this  example  were  calculated  from  the  example 
in  §  44.  The  values  ofCBl  and  B  D  here  found  differ  slightly 
from  those  obtained  before.  In  general,  the  triangle  D  B  P  is  oi 
better  form  for  accurate  calculation  than  the  triangle  A  D  B. 

46.  If  no  circumstance  determines  either  of  the  radii,  the  con- 
dition maybe  introduced,  that  the  common  tangent  shall  be  para> 
lei  to  the  line  joining  the  tangent  points. 

Problem.  Given  the  line  AB  —  a  (fig.  10),  which  unites  th^ 
fixed  tangent  points  A  and  B,  the  angle  I A  B  =  A,  and  the  angle 
ABI—B,to  find  the  radii  A  E  =  R  and  B  F  =  R'  of  a  com- 
pound curve,  having  the  common  tangent  D  G  parallel  to  A  B. 


COMPOUND   CURVES. 


33 


Solution.    Let  A  C  and  B  C  be  the  two  branches  of  the  required 
curve,  and  draw  the  chords  A  C  and  B  C.     These  chords  bisect 


Fig.  12. 


the  angles  A  and  B ;  for  the  angle 
and  the  angle  GBC  =  ^DGI=^ABL  Then  in  the  triangle 
A  CB  we  have  A  C :  A  B  =  sin.  ABC:  sin.  A  C  B.  But  A  CB  = 
180°  -(CAB+CBA)  =  180°  -  i  (A  +  B),  and  as  the  sine 
of  the  supplement  of  an  angle  is  the  same  as  the  sine  of  the 
angle  itself,  sin.  A  C  B  =  sin.  |  (A  +  B).  Therefore  A  C  :  a  = 

a  sin.  •$•  B 


sin.  £  B  :  sin.  |  (A  +  ^),  or  A  C  — 


manner  we  should  find  B  C  — 


sin.  £  (A  + 
'   — 


In  a  similar 


(§  82)  R  = 


C 


-, ,  and  R  = 
sin.  -J  A  i 

of  A  C  and  J?  (7  just  found. 


. 
sin. 

t  B  C 


—  ™.     Now  we  have 
+  1?) 

,  or,  substituting  the  values 


-,R  = 


$  a  sin.  |  J. 
sin.  ^  B  sin.  -J  ( J. 


34  CIRCULAR    CURVES. 

Example.    Given  a  =  950,  A  =  8°,  and  B  =  7°,  to  find  R  and 
R'.    Here 

i  a  =  475  2.676694 

i^  =  3°  30'  sin.  8.785675 


1.462369 

i  4  =  4°     sin.  8.843585 
+  -B)  =  7°  30'  sin.  9.115698 

7.959283 


^  =  3184.83  3.503086 

Transposing  these  same  logarithms  according  to  the  formula 
for  R  we  have 

i  a  =  475  2.676694 

•M=4°  sin.  8.843585 


1.520279 

1 5  =  3°  30'     sin.  8.785675 
|  (A  +  B)  =  7°  30'     sin.  9.115698 

7.901373 


R  =  4158.21  3.618906 

47.  Problem.  Given  the  line  AB  —  a  (fig.  12\  which  unites 
the  fixed  tangent  points  A  and  B,  and  the  tangents  AI=T  and 
B  I  =  T',  to  find  the  tangents  A  D  =  x  and  B  G  =  y  of  the  two 
branches  of  a  compound  curve,  having  its  common  tangent  I)  Q 
parallel  to  A  B. 

Solution.  Since  D  C  =  A  D  =  x,  and  C  &  =  B  G  =  y,  we  have 
D  G  =  x  +  y.  Then  the  similar  triangles  IDG  and  I  A  B  give 
ID  :  I  A  =  D  G  :  A  B,  or  T  —  x  :  T  =  x  +  y  :  a.  Therefore 
a  T  —  ax  =  Tx  +  Ty  (1).  Also  A  D  :  A  I  =  B  G  :  B  7,  or 
x\T—y\T.  Therefore  T  y  =  T1  x  (2).  Substituting  in  (1)  the 
value  of  Ty  in  (2),  we  have  a  T  —  ax  =  Tx  +  T'  x,  or  ax  + 
Tx  +  T'  x  =  a  T-, 

aT 


.  *  .X  = 


a+T+T" 


T'  x 
and,  since  from  (2),  y  —  — ~- , 


- 

y  ~  a  +  T  +  T' ' 


COMPOUND    CURVES. 


35 


The  intersection  points  D  and  G  and  the  common  tangent  point 
C  are  now  easily  obtained  on  the  ground,  and  the  radii  may  be 
found  by  the  usual  methods.  Or,  if  the  angles  I  A  B  =  A  and 
A  B  I  =  B  have  been  measured  or  calculated,  we  have  (§  5)  R  = 
x  cot.  -J-  A,  and  R'  =  y  cot.  £  B.  Substituting  the  values  of  a;  an  1 


y  found  above,  we  have  R  = 


a  + 


,  and  R  = 


a+  T  +  T>  • 


Example.  Given  a  =  500,  T  =  250,  and  T  =  290,  to  find  x 
and  y.  Here  a  +  T  +  T'  —  500  4-  250  4-  290  -  1040  ;  whence 
x  =  500  x  250  -*-  1040  =  120.19,  and  y  =  500  x  290  -f-  1040  = 
139.42. 


48.   Problem.     Given  the  tangents  A  I  =  T<  B I  =  T' ,  and 

the  angle  of  intersection  I,  to  unite,  the  tangent  points  A  and  B 


Fig.  13. 


*  The  radii  of  an  oval  of  given  length  and  breadth,  or  of  a  three-centre 
arch  of  given  span  and  rise,  may  also  be  found  from  these  formulae.  In 
these  cases  A  +  B  =  90°,  and  the  values  of  R  and  R'  may  be  reduced  to  R  = 
aT  aT' 


-  and  R'  = 


a  +  T'-T 

tion,  or  they  may  be  readily  calculated. 


These  values  admit  of  an  easy  construe- 


36  CIRCULAR   CURVES. 

(fig.  13)  by  a  compound  curve,  on  condition  that  the  two  branches 
shall  have  their  angles  of  intersection  IDG  and  I  G  D  equal. 

Solution.  Since  IDG  =  IGD  =  $I,  we  have  I  D  -  I  G. 
Represent  the  line  ID  =  I  G  by  x.  Then  if  the  perpendicular 
IH  be  let  fall  from  J,  we  have  (Tab.  X.  11)  DH=ID  cos.  IDG  = 
x  cos.  i  J,  and  D  G  =  2  x  cos.  i  I.  But  D  G  =  D  C  +  C  G  = 
AD+BG=T—  x  +  T'  —  x  =  T  +  T'  —  2  x.  Therefore 
2xcos.$I=  T  +  T'  —  2x,  or  2x  +  2xcos.il=  T  +  T  \  whence 

or  (T  b  x  ^ 

' 


_ 
1  +  cos. 


"    cos.2*/  ' 

The  tangents  AD  =  T  —  x  &ud  B  G  =  T'  —  x  are  now  readily 
found.  With  these  and  the  known  angles  of  intersection,  the  radii 
or  deflection  angles  may  be  found  (§  5  or  §  11).  This  method  an- 
swers very  well,  when  the  given  tangents  are  nearly  equal ;  but  in 
general  the  preceding  method  is  preferable. 

Example.  Given  ^=480,  T'=500,  and  J=18°,  to  find  x. 
Here 

J(5T+  r')  =  245  2.389166 

i/=4°30'     2  cos.  9.997318 

x  =  246.52  2.391848 

Then  A  D  =  480  -  246.52  =  233.48,  and  B  G  =  500  -  246.52  = 
253.48.  The  angle  of  intersection  for  both  branches  of  the  curve 
being  9°,  we  find  the  radii  A  E  =  233.48  cot.  4°  30'  =  2966.65,  and 
B  F  =  253.48  cot.  4°  30'  =  3220.77. 

ARTICLE  III. — TURNOUTS  AND  CROSSINGS. 

49.  The  turnouts  here  considered  are  of  three  kinds :  Those  in 
which  a  pair  of  rails  in  the  main  track  are  switched,  and  the  turn- 
out curve  is  made  tangent  to  the  switched  rails ;  those  in  which  a 
point  switch,  sometimes  called  a  split  switch,  is  employed,  to  one 
side  of  which,  when  thrown,  the  turnout  curve  is  made  tangent ; 
and  those  in  which  a  pair  of  rails  of  the  main  track  are  switched 
in  such  a  way  that  they  become  part  of  the  turnout  curve,  which 
thus  becomes  tangent  to  the  main  track.  The  problems  that  im- 
mediately follow  (§  50  to  §  64)  are  applicable  to  the  first  two  cases. 
Problems  relating  to  the  third  case  will  follow  (g  65  to  §  76). 


TURNOUT    FROM    STRAIGHT   MAIN    TRACK. 


37 


First  and  Second  Cases. 

50.  Let  A  B  (fig.  14)  represent  either  a  switched  rail,  or  the  side 
of  a  point  switch  when  thrown.     To  this  line  the  outer  rail  B  F 
of  the  turnout  is  tangent,  and  crosses  the  main  track  at  F.    The 
angle  G  F  M,  denoted  by  F,  is  called  the  frog  angle,  and  the  an- 
gle D  A  B,  denoted  by  S,  is  called  the  switch  angle.    The  gauge 
of  the  track  D  (7,  denoted  by  g,  and  the  distance  D  B,  called  the 
•^hrow,  denoted  by  d,  are  supposed  to  be  given.     The  distance 

A  B  =  Us  also  given,  whence  we  have  sin.  S  =  -7—5  =  7  •    If,  for 

A.  Jj       I 

example,  we  had  A  B  =  I  =  18,  and   d  =  .42,  we  should  have 

sin.  S  =  ~  =  .02333,  or  S  =  1°  20'. 
lo 

A.  Turnout  from  Straight  Main  Track. 

51.  Problem.     Given  the  radius  R  of  the  centre  line  of  a 
turnout  (fig.  14),  to  find  the  frog  angle  G  F  M  =  F  and  the  chord 


Solution.  Through  the  centre  E  draw  E  K  parallel  to  the 
main  track.  Draw  B  H  and  F  K  perpendicular  to  E  K,  and  join 
B  F.  Then,  since  E  F  is  perpendicular  to  F  M  and  F  K  is  per- 
pendicular to  F G,  the  angle  E FK  =  GFM=F-,  and  since 
E  B  and  B  H  are  respectively  perpendicular  to  A  B  and  A  D, 
the  angle  EBH-DAB  — S.  Now  the  triangle  EFK  gives 


38  CIRCULAR   CURVES, 

rr  JT" 

(Tab.  X.  2)  cos.  EFK  —  ^--^ .    But  E F,  the  radius  of  the  outer 


rail,  is  equal  to  R  +  \ g,   and  FK=CH  =  BH-BC- 
B  E  cos.  E  BH—  BC  =(R  +  \g}  cos.  S  —  (g  -  d).    Substituting 

™  ^  T^     (R  4-  -J-  g}  cos.  $  —  (g  —  d) 

these  values,  we  have  co$.EFK  =  —          y/         -^ — ^ z ,  or 

•R  +  ig 

cos.  .F  =  cos.  S  —  —- : — . 

-/t  +  kg 

From  this  formula  F  may  be  found  by  the  table  of  natural 
cosines.  To  adapt  it  to  calculation  by  logarithms,  we  may  con- 
sider g  —  d  to  be  equal  to  (g  —  d)  cos.  S,  which  will  lead  to  no 
material  error  since  g  —  d  is  very  small,  and  cos.  S  almost  equal 
to  unity.  The  value  of  cos.  F  then  becomes 


To  find  BF,  the  right  triangle  B  CF  gives  (Tab.  X.  9)  B  F  =* 

T>  rt 

But  B  C-g-d  and  the  angle  BFC=BFE  - 


siu.BFC' 

CFE  =  (90°  —\BEF)  —  (90°  -F)  =  F-$BEF.  But 
BEF  =  BLF—  E  B  L  =  F  —  S.  Therefore  B  F  C  =  F  — 
|  (F  —  S)  =  |  (F  4-  S).  Substituting  these  values  in  the  formula 
for  B  F,  we  have 

BF=   .     ?-*    . 


Example.  Given  g  =  4.7,  d  —  .42,  £  =  1°  20',  and  R  =  500,  to 
find  F  and  .B.F.  Here  nat.  cos.  S  =  .999729,  g  -  d  =  4.28, 
.#  +  ^  -  502.35,  and  4.28  -«-  502.35  =  .008520.  Therefore  nat. 
cos.  F  =  .999729  -  .008520  =  .991209,  which  gives  F  =  7°  36'  10". 
Next,  to  find  B  F, 

g-d  =  4.28  0.631444 

±(F  +  S)  =  4°  28'  5"     sin.  8.891555 

B  F  =  54.94  1.739889 

52.  Problem.  Given  the  frog  angle  G  F  M  =  F  (fig.  14),  to 
find  the  radius  R  of  the  centre  line  of  a  turnout,  and  the  chord 
BF. 

Solution.     From  the  preceding  solution  we  have   cos.  F  = 


TURNOUT   FROM   STRAIGHT    MAIN    TRACK.  39 

(g-d)  ^     Theref  ore  (£  +  }  g)  cos.  F=(R  + 


R  +  ig 
t  g)  cos.  S  —  (ff  —  d),  or 


For  calculation  by  logarithms  this  becomes  (Tab.  X.  29) 

^  E  +  ^g  =  sin.lt(F+S)sm.$(F-Sy 

Having  thus  found  R  +  \g,  we  find  R  by  subtracting  \g.    BF 
is  found,  as  in  the  preceding  problem,  by  the  formula 


Example.     Given  #  =  4.7,  d  =  .42,  S  =  1°  20',  and  F  =  7°,  to 
find  R.    Here 

I  (g  -  d)  =  2.14  0.330414 

+  S)  =  4°  10'    sin.  8.861283 
_  #)  -  2°  50'    sin.  8.693998 

7.555281 


R  +  !£  =  595.85  2.775133 

.-.^  =  593.5 

Frogs  on  some  roads  are  designated  by  numbers  denoting  the 
ratio  of  the  length  of  the  frog  to  its  width,  the  width  being  a  line 
drawn  across  the  widest  part  of  the  frog,  and  the  length  a  per- 
pendicular on  this  line  from  the  point  of  the  frog;  so  that  if  the 
number  of  the  frog  be  denoted  by  w,  we  shall  have 

cot.  \  F  =  2  n. 

Then  to  find  -J-  F  we  find  the  angle  whose  cotangent  is  double 
the  number  of  the  frog.  Thus  for  frog  number  7  we  look  for  the 
angle  whose  cotangent  is  14,  and  we  find  $  F  =  4°  5'  8".  The 
frog  angles  in  Tab.  V.  are  so  computed. 

53.  Problem.  To  find  mechanically  the  proper  position  of 
a  given  frog. 

Solution.  Denote  the  length  of  the  switch  rail  by  /,  the  length 
of  the  frog  by  /,  and  its  width  by  w.  From  B  as  a  centre  with  a 
radius  BH=:2l,  describe  on  the  ground  an  arc  tf -fiT/T  (fig.  15), 


40  CIRCULAR    CURVES. 

and  from  the  inside  of  the  rail  at  #  measure  @H=2d,  and  from 
H  measure  HK  such  that  HK\BH—^w  : /,  or  HK:  21  = 

%w  :/;  that  is,  H K  =  —^.     Then  a  straight  line  through  B  and 


the  point  K  will  strike  the  inside  of  the  other  rail  at  F,  the  place 
for  the  point  of  the  frog.  For  the  angle  HBKhas  been  made 
equal  to  i  F,  and  if  B  M  be  drawn  parallel  to  the  main  track,  the 
angle  MB  H  is  seen  to  be  equal  to  $S.  Therefore,  MBK  = 
B  F  C  =  i  (F  +  £),  and  this  was  shown  (§  50)  to  be  the  true  value 
of  BFC. 

54.  If  the  turnout  is  to  reverse,  and  become  parallel  to  the  main 
track,  the  problems  on  reversed  curves  already  given  will  in  gen- 
eral be  sufficient.  Thus,  if  the  tangent  points  of  the  required 
curve  are  fixed,  the  common  radius  may  be  found  by  §  40.  If  the 
tangent  point  at  the  switch  is  fixed,  and  the  common  radius  given, 
the  reversing  point  and  the  other  tangent  point  may  be  found  by 
§  37,  the  change  of  direction  of  the  two  tangents  being  here  equal 
to  S.  But  when  the  frog  angle  is  given,  or  determined  from  a 
given  first  radius,  and  the  point  of  the  frog  is  taken  as  the  revers- 
ing point,  the  radius  of  the  second  portion  may  be  found  by  the 
following  method. 

Problem,  (riven  the  frog  angle  F  and  the  distance  HB  = 
b  (fig.  16)  between  the  main  track  and  a  turnout,  to  find  the  radius 
R'  of  the  second  branch  of  the  turnout,  the  reversing  point  being 
taken  opposite  F,  the  point  of  the  frog. 

Solution.  Let  the  arc  F  B  be  the  inner  rail  of  the  second 
branch,  F  &  —  R '  —  \g  its  radius,  and  B  the  tangent  point  where 
the  turnout  becomes  parallel  to  the  main  track.  Now  since  the 
tangent  F  K  is  one  side  of  the  frog  produced,  the  angle  HF  K  — 


TURNOUT    FROM    STRAIGHT    MAIN    TRACK. 


41 


F,  and  since  the  angle  of  intersection  at  K  is  also  equal  to  Ft 
BFK--=ltF(§  2, II.) ;  whence  B FH=^F.    Then  (§  82)  FG  = 


Fig.  16. 


$BF 
sin.BFK 
X.9),orJ£^=^ 
have 


.    But  BF=- 


. 
sin.  Y  JP 


0111.  Y^-  sm.BFH 

.    Substituting  this  value  of  |  B  F,  we 


In  measuring  the  distance  HB  =  b,it  is  to  be  observed,  that 
the  widths  of  both  rails  must  be  included. 


Example.    Given  b  —  6.2  and  F  =  8°,  to  find  R  '.    Here 

i  6  =  3.1  0.491362 

i-F=4°  sin.  8.843585 


=  44.44  1.647777 

=4°  sin.  8.843585 


>  -%g  =  637.08  2.804192 

.-.#'±=639.43 


42  CIRCULAR   CURVES. 

B.  Crossings  on  Straight  Lines. 

55.  When  a  turnout  enters  a  parallel  main  track  by  a  second 
switch,  it  becomes  a  crossing.    As  the  switch  angle  is  the  same  on 
both  tracks,  a  crossing  on  a  straight  line  is  a  reversed  curve  be- 
tween parallel  tangents.     Let  HD  and  N  K  (fig.  17)  be  the  centre 
lines  of  two  parallel  tracks,  and  HA  and  .5  JTthe  direction  of  the 
switched  rails.    If  now  the  tangent  points  A  and  B  are  fixed,  the 
distance  A  B  =  a  may  be  measured,  and  also  the  perpendicular 
distance  B  P=  b  between  the  tangents  HP  and  B  K.    Then  the 
common  radius  of  the  crossing  AC  B  may  be  found  by  §  33  ;  or 
if  the  radius  of  one  part  of  the  crossing  is  fixed,  the  second  radius 
may  be  found  by  §  34.    But  if  both  frog  angles  are  given,  we  have 
the  two  radii  or  the  common  radius  of  a  crossing  given,  and  it 
will  then  be  necessary  to  determine  the  distance  A  B  between  the 
two  tangent  points. 

56.  Problem.     Given  the  perpendicular  distance  G  N  =  b 
(fig.  17)  between  the  centre  lines  of  two  parallel  tracks,  and  the 
radii  E  G  =  R  and  C  F  =  R  '  of  a  crossing,  to  find  the  chords  A  C 
and  B  C. 

Solution.  Draw  E  G  perpendicular  to  the  main  track,  and 
A  L,  CM,  and  B  L  '  parallel  to  it.  Denote  the  angle  A  E  C  by  E. 
Then,  since  the  angle  AEL  =  Aff6r  =  S,wQ  have  C  E  L  =  E  + 
S,  and  in  the  right  triangle  GEM  (Tab.  X.  2),  CEcos.  CEM= 
Rcos.(E  +  S)  =  EM=EL-LM.  But  EL  =  AEcos.A  EL 
=  E  cos.  S,  and  L  M  :  L'  M=  A  C  :  B  C.  Now  A  C  :  B  C  = 
EC:  CF=R:R'.  Therefore,  L  M  :  L'  M=  R  :  R\  or  L  M  : 
LM  +  L'  M=R\R  +  R'  ;  that  is,  LM  :  b  -2d  =  R:  R  +  R' 

whence  LM=     j_  ~     /.  .    Substituting  these  values  of  E  L  and 
L  H  in  the  equation  for  R  cos.  (E  +  S),  we  have  R  cos.  (E  +  S)  = 


.  '  .  cos.  (E  +  S)  =  cos.  S  -     —      - 
j-t  +  M 

Having  thus  found  E  +  S,  we  have  the  angle  E  and  also  its 
equal  CFB.    Then  (§  83) 


TURNOUT   FROM   CURVES. 


43 


We  have  also  AB  =  AC  +  B  C,  since  A  C  and  B  C  are  in  the 
same  straight  line  (§  32),  or  A  B  =  2  (R  +  R ')  sin.  J  E. 


Fig.  17. 


When  the  two  radii  are  equal,  the  same  formulae  apply  by  mak- 
ing R'  —  R.    In  this  case,  we  have 

cos.  (E  +  S)  =  cos.  S  - 


2  R 


Example.  Given  d  =  .42,  g  -  4.7,  S  =  1°  20',  b  =  11,  and  the 
angles  of  the  two  frogs  each  7°,  to  find  AC=BC=%AB.  The 
common  radius  R,  corresponding  to  F  =  7°,  is  found  (§  52)  to  be 
593.5.  Then  2  R  =  1187,  b-2d=  10.16.  and  10.16  -t-  1187  = 
.00856.  Therefore,  nat.  cos.  (E  4-  S)  =  .99973  -  .00856  =  .99117; 
whence  E  +  S  =  7°  37'  15".  Subtracting  S,  we  have  E  =  6°  17'  15". 
Next 

212  =  1187  3.074451 

i^^3°8'37^"     sin.  8.739106 


AC=  65.1 


1.813557 


C.  Turnout  from  Curves. 

57.  Problem.  #wm  the  radius  R  of  the  centre  line  of  the 
main  track  and  the  frog  angle  F,  to  determine  the  position  of  the 
frog  by  means  of  the  chord  B  F  (figs.  18  and  19),  and  to  find  the 
radius  R  of  the  centre  line  of  the  turnout. 

Solution.    I.  When  the  turnout  is  from  the  inside  of  the  curve 


44 


CIRCULAR    CURVES. 


(fig.  18).    Let  A  G  and  CF  be  the  rails  of  the  main  track,  A  B 
the  switch  rail,  and  the  arc  B  F  the  outer  rail  of  the  turnout, 


Fig.  18. 


crossing  the  inside  rail  of  the  main  track  at  F.  Then,  since  the 
angle  E  F K  has  its  sides  perpendicular  to  the  tangents  of  the 
two  curves  at  F,  it  is  equal  to  the  acute  angle  made  by  the  cross- 
ing rails,  that  \s>,EFK—F.  Also  E  B  L  =  S.  The  first  step  is 
to  find  the  angle  B  KF  denoted  by  K.  To  find  this  angle,  we 
have  in  the  triangle  BFK  (Tab.  X.  14)  BK+KF -.BE- 
EF =  tan.  i  (B  FE  +  FB  E}  :  tan.  £  ( B  F  E  -  F  B  E).  But 
B  E  =  R  +  \g  -  d,  and  E  F  =  R  -  $  g.  Therefore,  B  E  + 
KF-^R-d,  and  BE-EF  =  g-d.  Moreover,  BFE  = 
BFE  +  EFE=BFE  +  F,w&FBE=EBF-EBE  = 
B  F  E  -  S.  Therefore,  BFE-FBE=F+S.  Lastly, 
BFE  +  FBE—  180°  —  K.  Substituting  these  values  in  the 
preceding  proportion,  we  have  2  R  —  dig  —  d  =  tan.  (90°  — 


But  tan.  (90°  -  |  K)  =  cot.  £  Jf  =  - 


9-< 


g-d 


TURNOUT  FROM  CURVES.  45 

Next,  to  find  the  chord  B  F,  we  have,  in  the  triangle  B  F  G 
(Tab.  X.  12),  £f  =  S^nairF-     But  B  C  =  g  -  d,  and 

al  II.  Jj  J>    \j 

B  CF  =  180°  -  F  C  K  =  180°  -  (90°  -  ±  K)  =  90°  +  i  K,  or 
sin.  5  CF=cos.$K.   Moreover,  BFC  =  i(F  +  S)  ;  toiBFK  ~ 


Therefore,  B  F  K  —  FBK=2BFC.  But,  as  shown  above, 
BFK-FBK=F+  S.  Therefore,  2  B  F  C  =  F  +  £,  or 
BFC  =  $(F  +  S).  Substituting  these  values  in  the  expressior 
for  B  F,  we  have 

(g-d)GoS.$K 
' 


Lastly,  to  find  .#',  we  have  (§  %%)  R  +  \g  =  E  F  =   . 


. 
sin 

=  BLF-  EBL,an&  BLF  =  LFK  +  LKF  = 
F  +  K.    Therefore,  B  E  F  =  F  +  K  -  S,  and 


II.  When  the  turnout  is  from  the  outside  of  the  curve,  the  pre- 
ceding solution  requires  a  few  modifications.  In  the  present 
case,  the  angle  E  F  K1  =  F  (fig.  19)  and  E  B  L  =  S.  To  find 
K,  we  have  in  the  triangle  B  F  K,  KF+BK:  KF-BK- 
tan.  1  (F  B  K  +  B  F  K)  :  tan.  \(FBK-BFK).  But  K  F  = 
R  +  ±  g,  and  BK  —  R  —  $  g  +  d.  Therefore,  KF  +  BK  = 
2R  +  d,  and  KF-BK=g-d.  Moreover,  .F  £  JT  =  180°  - 
FBL  =  180°-  (EBF  -  E  B  L)  =  180°-^^^-  S),  and 
BFK=\W°  -  BFK'  =  180°  -  (B  F  E  +  E  F  K')  =  ISO0  - 
(EBF  +  F).  Therefore,  FB  K  -  BFK  =  F  +  S.  Lastly, 
FBK+  BFK=18Q°-K.  Substituting  these  values  in  the 
preceding  proportion,  we  have  2  ft  +  d  :  g  —  d  =  tan.  (90°  — 

*  JT)  :  tan.  *  (^  +  S),  or  tan.  (90°  -  *  K)  =  P  *  +  fl  ta 


But  tan.  (90°  -  £  JT)  =  cot.  }  K  = 


tan.  • 


Next,  to  find  B  F,  we  have,  in  the  triangle  B  F  C,  B  F  = 
=  9~d,  and  BCF=^°-^K,  or 


46  CIRCULAR   CURVES. 

sin.  B  C  F  =  cos.  •£  K.  Moreover,  B  F  C  =  $  (F  +  S)  ;  for 
BFK=  RFC  -  B  F  C,  and  FB  K=  KG  F  +  B  FC  = 
KFC+BFC.  Therefore,  FB  K-  BFK-  2  B  F  C.  But, 


as  shown  above,  FBK-BFK-F+S.  Therefore,  2BFC- 
F  +  S,  or  ^  F  C  =  |  (^  +  S).  Substituting  these  values  in  the 
expression  for  B  F,  we  have,  as  before, 

7?  F  -  (ff  -  <Q  cos-  -I  J5T* 

Z  ' 


Lastly,  to  find  R  ',  we  have  (§  82)  R1  +  $g  =  E  F- 


*  Since  $K  is  generally  very  small,  an  approximate  value  of  B  F  may  be 


obtained  by  making  cos.  IK=1.    This  gives  BF  = 
identical  with  the  formula  for  B  F  in  §  51. 


9-d 


sin.*  (-F  +  S) 


,  which  is 


TURNOUT  FROM  CURVES.  47 


nd  BLF-LFK-LKF^ 

F-  K.    Therefore,  B  E  F  =  F  -  K  -  S,  and 

$BF 

KF-K-S)' 

Example.  Given  g  =  4.7,  d  =  .42,  S  =  1°  20',  R  =  4583.75,  and 
F=  7°,  to  find  the  chord  B  F  and  the  radius  R'  of  a  turnout  from 
the  outside  of  the  curve.  Here 

g-d  =  4.28  0.631444        0.631444 

2R  +  d  =  9167.92  3.962271 

l(F  +  S)  =  4°  10'         tan.  8.862433  sin.  8.861283 

2.824704        1.770161 


£"=  22'  1.8"  tan.  7.806740  cos.  9.999991 


^^=58.905  1.770152 

2  0.301030 

1  (F—  K-  S)  =  2°  27'  58.2"         sin.  8.633766 

8.934796 


E  +  ig  =  684.47  2.835356 

.-.  R'  =  682.12 

58.  Problem.     To  find  mechanically  the  proper  position  of 
a  given  frog. 

Solution.  The  method  here  is  similar  to  that  already  given, 
when  the  turnout  is  from  a  straight  line  (§  53).  Draw  B  M  (figs. 
18  and  19)  parallel  to  F C,  and  we  have  FBM  =  BFC  =  i(F  + 
S),  as  just  shown  (§-  57).  This  angle  is  to  be  laid  off  from  B  M\ 
but  as  F  is  the  point  to  be  found,  the  chord  F  C  can  be  only  esti- 
mated at  first,  and  B  M  taken  parallel  to  it,  from  which  the  angle 
%  (F  +  S)  may  be  laid  off  by  the  method  of  §  53.  In  this  case, 
however,  the  first  measure  on  the  arc  is  d,  and  not  2  d ;  since  we 
have  here  to  start  from  B  Jf,  and  not  from  the  rail.  Having  thus 
determined  the  point  F  approximately,  B  M  may  be  laid  off  more 
accurately,  and  F  found  anew. 

59.  Problem.     Given  the  position  of  a  frog  by  means  of  the 
chord  B  F  (figs.  14,  18,  and  19),  to  determine  the  frog  angle  F. 

a  —  d 

Solution.    The  formula  BF  =  - — f-= ^-,  which  is  exact 

sm.i(F+  S) 


48  CIRCULAR   CURVES. 

on  straight  lines  (§  51),  and  near  enough  on  ordinary  curves  (§  57. 
note),  gives 


By  this  formula  $(F  +  S)  may  be  found,  and  consequently  F. 

60.  Problem.  Given  the  radius  R  of  the  centre  line  of  the 
main  track,  and  the  radius  R  '  of  the  centre  line  of  a  turnout,  to 
find  the  frog  angle  F,  and  the  chord  B  F  (figs.  18  and  19). 

Solution.  I.  When  the  turnout  is  from  the  inside  of  the  curve 
(fig.  18).  In  the  triangle  B  E  K  find  the  angle  B  E  K  and  the 
side  E  K.  For  this  purpose  we  have  B  E  =  R'  +  \g,  B  K  — 
R  +  %g  _  d,  and  the  included  angle  E  B  K  '=  S.  Then  in  the 
triangle  EFKwe  have  E  K,  as  just  found,  E  F  =  R'  +  ±g,  and 
FK  =  R  —  i  g.  The  frog  angle  E  F  K  =  F  may,  therefore,  be 
found  by  formula  15,  Tab.  X.,  which  gives 


8  (s  -  a) 

where  s  is  the  half  sum  of  the  three  sides,  a  the  side  E  K,  and  6 
and  c  the  remaining  sides. 

Find  also  in  the  triangle  E  F  K  the  angle  F  E  K,  and  we  have 
the  angle  BEF=BEK— FEK.  Then  in  the  triangle 
BEF we  have  (§83) 

jjy  B  F=2(R'  +  i#)sin.  %  B  E  F* 

II.  When  the  turnout  is  from  the  outside  of  the  curve  (fig.  19). 
In  the  triangle  B  E  K  find  the  angle  B  E  K  and  the  side  E  K. 
For  this  purpose  we  have  B  E  =  R'  +  ig,  B  K  =  R  —  i  g  +  d, 
and  the  included  angle  E  B  K  =  180°  —  S.  Then  in  the  triangle 
E  F  K  we  have  E  K,  as  just  found,  E  F  =  R '  +  \g,  and  F  K  — 
R  +  \g.  The  angle  E  FKm&y,  therefore,  be  found  by  formula 


15,  Tab.  X.,  which  gives  tan.  i  E  F  K  =  \/~         ~     .    But 

s  (s  —  a) 


*  The  value  of  B  F  may  be  more  easily  found  by  the  approximate  formula 

BF—  - — 7-^ — r-,  and  generally  with  sufficient  accuracy.     See  note  to 

sin.  i  V-P  +  £> ) 
§  57.    This  remark  applies  also  to  B  F  in  the  second  part  of  this  solution. 


TURNOUT  FROM  CURVES.  49 

the  angle  E  F  K'  =  F  =  180°  -  E  F  K.    Therefore  J  F  -  90°  - 
\EFK,  and  cot.  \F-  tan.  |  E  FK\ 


JE^P"  .       .  Wu.  ^  ^      -|/  ,  , 

o  ^o  ~~  U) 

where  s  is  the  half  sum  of  the  three  sides,  a  the  side  E  K,  and  I 
and  c  the  remaining  sides. 

^\'n^  also  in  the  triangle  E  F  K  the  angle  F  E  K,  and  we  have 
the  angle  BEF=FEK—BEK.  Then  in  the  triangle  B  E  F 
we  have  (§  83) 

BF  =  2(R' 


Example.    Given  g  =  4.7,  d  =  .43,  S  =  1°  20',  R  =  4583.75,  and 


jK'  =  682.12,  to  find  F  and  the  chord  B  F  of  &  turnout  from  the 
outside  of  the  curve.    Here  in  the  triangle  B  E  K  (fig.  19)  we  have 
5 


50  CIRCULAR   CURVES. 


W  +  ig  =  684.47,  B  K  =  R-±g  +  d  =  4581.82,  and  the 
angles  B  E  K  +  B  K  E  =  S  =1°  2V  '.    Then 

B  K  -  B  E  =  3897.35  3.590769 

i  (B  E  K  +  B  KE  )  =  40'  tan.  8.065806 


1.656575 
B  K  +  B  E  =  5266.29  3.721505 


%(BEK-  BKE}*  =  29.6029'      tan.  7.935070 
.-.BEK=\°  9.6029' 


EK  is  now  found  by  the  formula 

sm.  B  E  K 

log.  E  K=  log.  4581.82  +  log.  sin.  178°  40'  —  log.  sin.  1°  9.6029'  = 

3.721491,  whence  EK=  5266.12. 
Then  to  find  F,  we  have  in  the  triangle  E  F  K,  s  =  \  (5266.12  + 

684.47  +  4586.10)  =  5268.34,  s  -  a  =  2.22,  s  -  b  =  4583.87,  and 

8  -  o  =  682.24. 

a  -  ft  =  4583.87  3.661233 

s  -  c  =  682.24  2.833937 


6.495170 

s  =  5268.34   3.721674 
5  -  a  =  2.22      0.346353 


4.068027 

2)2.427143 

i^=3°30'  cot.  1.213571 

.  • .  F  =  r 

To  find  F  E  K,  we  have  s  as  before,  but  as  a  is  here  the  side 
F  K  opposite  the  angle  sought,  we  have  s  —  a  =  682.24,  s  —  b  = 
4583.87,  and  s  —  c  =  2.22.  Then  by  means  of  the  logarithms  just 
used,  we  find  |  FEK-  3°  2'  45".  Subtracting  | B  EK-  34'  48", 
we  have  $BEF  =  2°2T  57".  Lastly,  B  F  =  1368.94  sin.  2°  27' 
57"  =  58.897.  _ 

The  formula  B  F  =  - — £-^ ^-  (§  57,  note)  would  give  B  F  = 

58.906,  and  this  value  is  even  nearer  the  truth  than  that  just  found, 
owing,  however,  to  no  error  in  the  formulae,  but  to  inaccuracies 
incident  to  the  calculation. 

*  This  angle  and  the  sine  of  1°  9.6029'  below,  are  found  by  the  method 
given  in  connection  with  Table  XV.  If  the  ordinary  interpolations  had 
been  used,  we  should  have  found  F=7°  7',  whereas  it  should  be  7°,  since 
this  example  is  the  converse  of  that  in  §  57. 


TURNOUT  FROM  CURVES. 


51 


61.  If  the  turnout  is  to  reverse,  in  order  to  join  a  track  parallel 
to  the  main  track,  as  A  C  B  (fig.  20),  it  will  be  necessary  to  deter- 
mine the  reversing  points  C  and  B.    These  points  will  be  deter- 
mined, if  we  find  the  angles  A  E  C  and  B  F  C,  and  the  chords 
A  G  and  C  B. 

62.  Problem.     Given  the  radius  DK=R  (fig.  SO)  of  the 
centre  line  of  the  main  track,  the  common  radius  EC—  C  F  — 


H '  of  the  centre  line  of  a  turnout,  and  the  distance  B  G  =  b  be- 
tween the  centre  lines  of  the  parallel  tracks,  to  find  the  central 
angles  AEG  and  B  F  C  and  the  chords  A  C  and  B  C. 

Solution.  In  the  triangle  A  E  K find  the  angle  A  E  K  and  the 
side  E  K.  For  this  purpose  we  have  AE=R',AK=R  —  d, 
and  the  included  angle  E  A  K  —  S.  Or,  if  the  frog  angle  has 
been  previously  calculated  by  §  60,  the  values  of  A  E  K  and  E  K 
are  already  known.* 

Find  in  the  triangle  E  F  K  the  angles  E  F  K  and  F  E  K.  For 
this  purpose  we  have  E  K,  as  just  found,  E  F  =  2  R',  and  F  K  = 


*  The  triangle  A  EK  does  not  correspond  precisely  with  B  E  Kin  §  60,  A 
being  on  the  centre  line  and  B  on  the  outer  rail ;  but  the  difference  is  too 
slight  to  affect  the  calculations. 


52  CIRCULAR   CURVES. 

R  +  R'  —  l.      Then  AEC=AEK— FEK,   and 
EFK.    Lastly  (§83), 

B@P"    AC=2Rsm.iAEC,     CB  =  2R'si 


This  solution,  with  a  few  obvious  modifications,  will  apply, 
when  the  turnout  is  from  the  outside  of  a  curve. 


D.  Crossings  on  Curves. 

63.  When  a  turnout  enters  a  parallel  main  track  by  a  second 
switch,  it  becomes  a  crossing.  Then  if  the  tangent  points  A  and 
B  (fig.  21)  are  fixed,  the  distance  A  B  must  be  measured,  and  also 


Fig.  21. 


the  angles  which  A  B  makes  with  the  tangents  at  A  and  B.  The 
common  radius  of  the  crossing  may  then  be  found  by  §  40 ;  or  if 
one  radius  of  the  crossing  is  given,  the  other  may  be  found  by 
§  38.  But  if  one  tangent  point  A  is  fixed,  and  the  common  radius 
of  the  crossing  is  given,  it  will  be  necessary  to  determine  the  re- 
versing point  C  and  the  tangent  point  B.  These  points  will  be 
determined,  if  we  find  the  angles  AEG  and  B  F  (7,  and  the 
chords^.  C  and  C  B. 


TURNOUTS  TANGENT  TO  MAIN  TRACK.          53 

64.  Problem.  Given  the  radius  DK=R  (fig.  21)  of  the 
centre  line  of  the  main  track,  the  common  radius  E  C  =  C  F  = 
R '  of  the  centre  line  of  a  crossing,  and  the  distance  D  6r  =  b  be- 
tween the  centre  lines  of  the  parallel  tracks,  to  find  the  central 
angles  AEG  and  B  F  C  and  the  chords  A  C  and  C  B. 

Solution.  In  the  triangle  A  E  K  find  the  angle  A  E  K  and  the 
side  E  K.  For  this  purpose  we  have  AE  =  R',AK=R  —  d, 
and  the  included  angle  E  A  K—  S. 

Find  in  the  triangle  B  F  K  the  angle  B  F  K  and  the  side  F  K. 
For  this  purpose  we  have  BF=R',BK=R  —  b  +  d,  and  the 
included  angle  FBK=  180°  -  S. 

Find  in  the  triangle  E  F  K  the  angles  F  E  K  and  E  F  K.  For 
this  purpose  we  have  E  K  and  F  K  as  just  found,  and  E F  =  2R'. 
ThenAEC  =  AEK-FEK,  and  B  F C =E  F K -  B  F K. 
Lastly  (§  83), 


Third  Case. 
Turnouts  Tangent  to  Main  Track. 

65.  In  this  case  a  pair  of  rails  of  the  main  track  are  switched 
in  such  a  way  that  they  become  parts  of  the  turnout  curve.  Their 
length  in  relation  to  R,  the  radius  of  the  turnout,  must  be  deter- 
mined. Denote  their  length  by  I  and  the  "  throw  "  by  d.  Then 
on  the  centre  line  d  is  the  tangent  offset  of  a  curve  of  radius  R. 
By  §  18  this  offset  or  deflection  is  equal  to  the  square  of  the  chord 

I9 
divided  by  twice  the  radius,  or  d  =  £-= ; 

.-.1=  \/2~Rd. 

By  this  formula  column  I  in  Tab.  V.  is  calculated. 

A  switch-rail  may  be  made  to  take  the  proper  curve  in  the  fol- 
lowing manner :  Suppose  the  length  of  the  switch-rail,  as  calcu- 
lated above,  to  be  20  feet.  A  rail  30  feet  in  length  is,  for  10  feet 
back  from  the  tangent  point,  spiked  down,  or  otherwise  securely 
fastened  on  the  main  track,  leaving  20  feet  free  for  the  switch-rail. 
The  free  end  being  thrown  in  the  usual  way,  a  curve  is  formed, 
which,  however,  is  not  a  circular  curve,  but  an  elastic  curve.  The 
inclination  at  the  free  end,  in  the  case  supposed,  would  be  about 


54  CIRCULAR   CURVES. 

three-fourths  of  that  of  the  circular  curve  that  meets  it.  If  it  be 
desired  to  make  the  two  inclinations  equal,  so  that  the  two  curves 
shall  be  tangent  to  each  other,  the  switch-rail  should  be  only 
three-fourths  of  the  calculated  length  of  I.  The  switch-rail  may, 
however,  be  made  to  take  a  circular  form  by  suitable  stops  at- 
tached to  the  sleepers.  The  full  length,  as  calculated  above,  will 
then,  of  course,  remain  free.  The  offsets  from  the  tangent  to  the 
stops  will  be  to  d  as  the  squares  of  the  distances  from  the  tan- 
gent point  are  to  Z*. 


A.  Turnout  from  Straight  Lines. 

66.  Problem.  Given  the  radius  E  of  the  centre  line  of  a 
turnout,  and  the  gauge  B  G  ==  g  (fig.  22\  to  find  the  frog  angle 
GFM=F,and  the  chord  B  F. 

Solution.  The  angle  C  E  F,  having  its  sides  perpendicular  to 
G  F  and  F  M,  is  equal  to  G  F  M  =  F.  In  the  triangle  CEFwe 

1    1 
have  cos.  CE  F= 


Draw  ED  perpendicular  to  BF.  Then,  from  the  similar  tri- 
angles B  F  C  and  B  E  D,  we  have  the  angle  BFC=BED= 
|  F.  Therefore,  B  F  sin.  $F 


67.  Problem.  Given  the  frog  angle  GF  M=F  (fig.  22), 
and  the  gauge  BC  =  g,to  find  the  radius  R  of  the  centre  line  of  a 
turnout,  and  the  chord  B  F. 

Solution.    From  the  preceding  problem  we  have 

•v-Saa—  Z>    17 9 

^  ".^irrr 

In  the  triangle  B  E  D  we  have  B  E  sin.  B  E  D  =  |  B  F,  or 

(R  +  $g)sm.$F 


TURNOUT   FROM    STRAIGHT   LINES. 


DO 


To  put  R  in  another  form,  substitute  for  B  F  its  value  above, 
and  transfer  %g  to  the  second  member.     We  then  have  R  = 

Iff 

— 


If  now  the  frog  angle  F  is  expressed  by  means  of  the  ratio  n  of 
the  length  to  the  breadth  of  the  frog,  as  explained  in  §  52,  we 


have  cot.  J  F  =  2  n,  and,  substituting  this  value  in  the  expression 
for  R,  we  have 


By  the  formulae  of  this  section  the  values  of  F,  B  F,  and  R  in 
Table  V.  are  calculated. 

68.  A  ready  way  of  locating  the  turnout  curve  is  to  locate  the 
outer  rail  first  by  stretching  a  cord  from  B  to  F,  and  from  it  fix- 
ing the  curve  by  ordinates  at  the  centre  and  at  the  quarter  points. 
The  middle  ordinate  m  may  be  taken  in  all  cases  =  \g.  For 

-  ,  and  putting  in  the  value  of  R  +  |  g  above, 


(§26),ro  = 

and  reducing,  we  have  m  =  £  B  Fsin.  $  F  =  ±  g.  For  g  =  4.708, 
m  =  1.177.  At  the  quarter  points  the  ordinates  will  be  f  m  = 
0.883.  The  inner  rail  is  then  located  by  the  gauge. 

69.  If  the  turnout  is  to  reverse  and  become  parallel  to  the  main 
track,  the  formulae  of  §  53  apply  here  also. 


56 


CIRCULAR   CURVES. 


B.  Crossings  on  Straight  Lines. 

70.  When  a  turnout  enters  a  parallel  main  track  by  a  second 
curve,  it  becomes  a  crossing,  and  the  two  curves  form  a  reversed 
curve  between  parallel  tangents.  The  problems  that  arise  here 
have  been  solved  already  (§§  33-36). 


C.  Turnout  from  Curves. 

71.  Problem.  Given  the  radius  R  of  the  centre  line  of  the 
main  track  and  the  frog  angle  F,  to  determine  the  position  of  the 
frog  ~by  means  of  the  chord  B  F  (figs.  23  and  ££),  and  to  find  the 
radius  R  of  the  centre  line  of  the  turnout. 

Solution.  I.  Turnout  from  the  inside  of  the  curve  of  the  main 
track.  Let  B  G  and  C  F  (fig.  23)  be  the  rails  of  the  main  track,  and 
the  arc  B  F  the  outer  rail  of  the  turnout,  crossing  the  inner  rail  of 


Fig.  23. 


the  main  track  at  F.  Then,  since  the  angle  EFK  has  its  sides 
perpendicular  to  the  tangents  of  the  two  curves  at  F,  it  is  equal 
to  the  acute  angle  made  by  the  crossing  rails;  that  is,  E F  K  =  F. 


TURNOUT  FROM  CURVES.  57 

The  first  step  is  to  find  the  angle  B  K  F  denoted  by  K.  To  find 
this  angle,  we  have  in  the  triangle  BFK  (Tab.  X.,  14)  tan.  k 

mFK     irnin      (BK-KJ?)te*.i(BFK+FBK) 
(±tJ<  K-  ±  ±S  K)  -  -  BK+KF 

BK-  KF=BK  -  CK  =  g,  and  BK  +  KF  -  2  R.  Also, 
tan.  %(BFK  +  FBK}  =  tan.  i  (180°  -  K}  =  tan.  (90°  -  i  K)  = 
cot.^K,  and  BFK-FBK=BFK-BFE  =  F.  Substi- 

tuting these  values,  we  have  tan.  •£  F  =  —  ^-^  —  —  2  R  tan  ±  K1 
or  2  R  tan.  |  F  tan.  %K=g\ 

•   tan  lK- 
•    •  tan.  •$  JL  — 


75  —        -   T->  . 

if,  by  the  notation  of  §  52,  we  put  cot.  J  F  =  2  n. 
To  find  the  chord  B  F,  we  have  in  the  triangle  B  F  C,  B  F  — 

But  B  C  =  g,  and  sin.  B  CF  =  sin.  F  CK  = 

cos.  i  K.  Moreover,  B  F  C  =  i  F.  ~ForBFK=KFC  + 
BFC,KH&FBK=KCF-BFC  =  KFC-BFC.  There- 
f  ore,  by  subtraction,  BFK-FBK=2BFC.  But,  as  shown 
above,  BFK  —  FBK=F.  Therefore  B  F  C  —  i  F.  Sub- 
stituting these  values  in  the  expression  for  B  F,  we  have 

-9  cos,  j  K* 

~ 


Lastly,  to  find  J2',  we  have  in  the  triangle  B  E  F,  EFsm. 
%BEF=%BF.    But  EF=R'  +  \g,  and  the  exterior  angle 


II.  Turnout  from  the  outside  of  the  curve  of  the  main  track. 
Let  J5  Gr  and  C  jF7  (fig.  24)  be  the  rails  of  the  main  track,  and  the 
arc  B  F  the  outer  rail  of  the  turnout,  crossing  the  outer  rail  of  the 
main  track  at  F.  The  frog  angle  F  is  now  represented  by  the 
angle  E  F  K'  .  The  first  step  is  to  find  the  angle  B  K  F,  denoted 

*  Since  i  Kis  generally  very  small,  an  approximate  value  of  B  F  may  be 

obtained  by  making  cos.  \K=  1,  whence  BF—    .  g.  CT,  which  ia  identical 

sin.  5-  _r 

with  the  formula  for  B  F  in  §  66.    This  remark  applies  also  to  B  F  in  the 
second  part  of  this  solution. 


58 


CIRCULAR   CURVES. 


by  K.    To  find  this  angle,  we  have  in  the  triangle  B  F  K  (Tab.  X.. 

BFK} 


14),  t 


-  BFK)  = 


EutKF-BJR:=g,<dudKF  +  BK-2R.  Also,  tan. 

B  F  K)  =  tan.  |  (180°  -  K)  =  tan.  (90°  -  i  K)  =  cot.  £  K  and 


Fig.  24. 


FBK-BFK  =  (180°  -FBE)-  (180°  -  5jPJ5T')  =BFK'- 
FBE  =  BFK'  —  BFE  =  F.     Substituting  these  values,  we 


have    tan.} 
tan.  %K  =  g. 


^  tan.  J  . 


or    2  ^  tan. 


.      .  tail,  -g-  _£i  —    —  =5 -  —  -=r-  , 

JLli  J\, 

if,  by  the  notation  of  §  52,  we  put  cot.  \  F  =  2  n. 
To  find  the  chord  B  F,  we  have  in  the  triangle  B  F  C,  B  F  — 

B  C"sin  B  C  F 

.    But  BC  —  g,  and  sin. #67^=  sin. (90°  —  \K}  — 


sin.  . 

cos.  J  AT.   Moreover,  5^^7=1^.    For  BFK=  KFC-  BFC, 
and  FBK=  KCF  +  BFC^KFC  +  BFC.    Therefore,  ty 


TURNOUT  FROM  CURVES. 


59 


subtraction,  F  B  K  '-  B  F  K  =2  B  F  C.    But,  as  shown  above, 
FBK-BFK=F.    Substituting  these  values,  we  have 
F      g  cos,  j  K 
~-  ' 


Lastly,  to  find  72',  we  have  in  the  triangle  BE  F,  JZ  Fsin.  $ 
\BF.    But  EF=R'  +  |£,  and  the  angle  BEF- 
E  F  K  —  E  K.  _P  =  _r  —  JL,  , 


Example.  Given  g  =  4.708,  R  -  1910.08,  and  F  =  7°  9'  10",  to 
find  the  chord  B  F  and  the  radius  R  '  of  a  turnout  from  the  inside 
of  the  curve  (fig.  23). 

To  find  4  Ki  $g  =  2.354  0.371806 

i^=3°34'35"  cot.  1.204115 


To  find  BF: 


R  =  1910.08 


g  =  4.708 


1.575921 
3.281051 

tan.  8.294870 

0.672836 
cos.  9.999915 


0.672751 
i^=3°34'35"    sin.  8. 795038 

B  F  =  75.46 


To  find  R1: 


=  37.73 


R'  +  ig  =  459.87 
.-.^'=457.52 


1.877713 

1.576687 
sin.  8.914051 


2.662636 


72.  Problem.  Given  the  radius  R  of  the  centre  line  of  the 
main  track  and  the  radius  R'  of  the  centre  line  of  a  turnout,  to 
find  the  frog  angle  F,  and  the  chord  B  F  (figs.  23  and  24). 

Solution.  I.  Turnout  from  the  inside  of  the  curve  of  the  main 
track.  In  the  triangle  E  F  K  (fig.  23)  we  have  given  the  sides 
EK=R-R',EF=R'  +  ltg,  and  FK=R-^g,  to  find 
the  angle  E  F K  =  F.  By  formula  15,  Tab.  X.,  tan.  \F  = 


8(8  —  0) 


• ,  where  s  is  the  half  sum  of  the  three  sides,  a  the 


60 


CIRCULAR   CURVES. 


side  E  K  opposite  the  angle  sought,  and  b  and  c  the  remaining 
sides.     Therefore,  s  =  \(E K  +  E F  +  F K}  =  R,  s-a=s- 


Fig.  23. 


=R  —  R'  —  %g,  and  s  —  c  =  s  — 


=  %g.    Substituting  these  values,  we  have 


tan.  A  F  = 


R  x  R' 


By  §  71,  JBF=^p^4j=  where  iJf  is  the  angle  DKF. 
sin.  -g-  Jj 

When  F  has  been  found,  |7Tmay  be  found  by  the  formula  for 
tan.  ^  K  in  §  71 ;  but,  generally,  -J-  K  is  so  small  that  we  may  put 
cos.  i  K=  1,  and  we  have 

.   ^  ET,  nearly. 


DOT 

II.  Turnout  from  the  outside  of  the  curve  of  the  main  track. 
In  the  triangle  E  F  K  (fig.  24)  we  have  given  the  sides  E  K  = 
R  +  R',  E F=  R'  +  \g,  and  FK=R  +  $g,  to  find  the  angle 
E  F  K,  the  supplement  of  the  angle  E  F  K\  which  now  repre- 
sents the  frog  angle  F.  By  formula  15,  Tab.  X.,  tan.  £ EFK  — 

(s  —  o)  (s  —  c^  ^  wnere  s  is  the  half  sum  of  the  three  sides,  a  the 
s(s  —  a) 


TMKNOUT  FROM  CURVES. 


61 


side  E  K  opposite  the  angle  sought,  and  b  and  c  the  remaining 
sides.    Therefore  «  =  %(E K  +  EF  +  F K)  =  R  +  R'  +lg,  s  - 


Fig.  24. 


R'.    Substituting  these  values,  we  have  tan. \EF K=.  < 
R  x  R' 


R' 


•••tan'^V(JL±OT 


By 


where 


is  the  angle 


When  J?7  has  been  found,  |  -ff'  may  be  found  by  the  formula  for 
tan.  \K  in  §  71 ;  but,  generally,  -^^Tis  so  small  that  we  may  put 
cos.  \  K  —  1,  and  we  have 


BF=   .  g   .-,,  nearly, 
sin.  i  F 


CIRCULAR   CURVES. 


73.  If  the  turnout  is  to  reverse  in  order  to  join  a  track  parallel 
to  the  main  track,  as  A  C  B  (fig.  25),  it  will  be  necessary  to  deter- 
mine the  reversing  points  C  and  B.  These  points  will  be  deter- 
mined, if  we  find  the  angles  AEG  and  B  F C,  and  the  chords 
A  C  and  B  C. 


74.  Problem.  Given  the  radius  A  K  =  R  (fig.  25)  of  the 
centre  line  of  the  main  track,  the  common  radius  E  C  = 
C  F  =  R'  of  the  centre  line  of  a  turnout,  and  the  distance 
B  G  —  b  between  the  centre  lines  of  the  parallel  tracks,  to  find 
the  central  angles  AEG  and  B  F  C,  and  the  chords  A  C  and 
BG. 

Fig.  25. 


Solution.  In  the  triangle  E  FK  find  the  angles  E  F  K  and 
F  E  K.  For  this  purpose  we  have  the  sides  of  the  triangle  given 
—namely,  EK-R-R',  EF-^R\  and  FK=R  +  R'  ~b. 

Then,  by  formula  15,  Tab.  X.,  tan.i  A  =  */(*-*>)(*-<*)    where 

s  (s  —  a) 

s  is  the  half  sum  of  the  three  sides,  a  the  side  opposite  the  angle 
sought,  here  denoted  by  A,  and  ~b  and  c  the  remaining  sides, 
Putting  FJZKfor  A,  and  FKfor  a,  we  shall  have  an  expression 
for  t&r\.%FEK=  tan. | (180°  —  AEG)  =  coi.^AEC,  and  put- 
ting E  F  K  for  A  and  E  K  for  a,  we  shall  have  an  expression  for 


DOUBLE   TURNOUTS.  63 

tan.  \EFK-  tan.  \BFC.    Making  the  proper  substitutions  in 
the  formula  for  tan.^J.,  we  shall  have 


Having  found  A  E  C  and  B  F  C,  we  have  the  chords 


This  solution,  with  a  few  obvious  modifications,  will  apply  when 
the  turnout  is  from  the  outside  of  the  curve. 

75.  Problem.     Given  the  position  of  a  frog  by  means  of  the 
vhord  B  F  (figs.  22,  23,  and  24),  to  find  the  frog  angle  F. 

Solution.     The  formula  BF  =    .    g,  CT,   which  is  exact  on 

sin.  •£  F 

straight  lines  (§  66).  and  near  enough  on  ordinary  curves  (§  71, 
note),  gives 


D.  Crossings  on  Curves. 

76.  When  a  turnout  enters  a  parallel  main  track  by  a  second 
switch,  it  becomes  a  crossing.     Then,  if  the  tangent  points  A  and 
B  (fig.  25)  are  fixed,  the  distance  A  B  must  be  measured,  and  also 
the  angles  made  by  AB  with  the  tangents  at  A  and  B.    The 
common  radius  of  the  crossing  may  then  be  found  by  §  40,  or  if 
one  radius  of  the  crossing  is  given,  the  other  may  be  found  by 
§  38.    But  if  one  tangent  point  A  is  fixed,  and  the  common  radius 
of  the  crossing  is  given,  the  reversing  point  C  and  the  second 
tangent  point  B  may  be  found  by  the  problem  of  §  74. 

E.  Double  Turnouts. 

77.  The  cases  that  arise  when  two  turnouts  start  from  the  same 
point  on  the  main  track  fall  under  problems  already  solved. 


64 


CIRCULAR   CURVES. 


Thus  when  the  outer  rails  of  two  turnouts,  as  B  C F  and  B'  C  F' 
(fig.  26),  turn  opposite  ways,  B'  C  F'  may  be  treated  as  a  turnout 
from  the  outside  of  the  inner  rail  B '  D  of  B  C  F.  Then  if  the 
frog  angle  at  C  is  given,  the  radius  of  B'  OF'  may  be  found  by 


Fig.  26. 


§  57  or  §  71,  or  if  the  radius  of  B '  C  F'  is  given,  the  frog  angle  at 
C  may  be  found  by  §  60  or  §  72, 

Or,  the  third  frog  may  be  placed  with  its  point  in  the  centre 
line  of  the  main  track,  and  its  angle  may  be  taken  as  made  up  of 
two  angles,  F\  and  F<*,  one  on  each  side  of  said  centre  line,  as  in 
figure  26.  On  a  straight  main  track  the  two  turnouts  would  in 
general  be  symmetrical,  and  FI  be  equal  to  F^.  On  a  curved 
main  track  these  partial  angles  may  be  equal  or  unequal.  All 
the  relations  between  the  radii  and  the  frog  angles  concerned  may 
be  determined  by  previous  problems,  substituting  \g  for  g  as  the 
distance  of  the  line  C  H  from  either  rail.  Thus  in  the  figure  the 
radius  of  B  C  and  the  partial  frog  angle  FI  depend  on  each  other, 
so  also  do  the  radius  of  B'  C  and  the  partial  frog  angle  F* 
When  one  of  the  chords,  as  B  C,  is  fixed  in  length,  the  length  of 
the  other,  B'  C,  is  also  fixed,  whether  equal  to  B  C  on  straight 
lines  or  different  on  curves.  The  partial  frog  angle  F^  being  de- 


DOUBLE   TURNOUTS. 


65 


pendent  on  the  length  of  B '  C,  is  found  by  §  59  or  §  75,  and  from 
it  the  radius  of  the  curve  B'  C  is  calculated. 

When  either  curve  beyond  (7,  as  C  F,  is  not  a  continuation 
of  the  curve  B  C,  the  relation  between  its  radius  and  the  frog 
angle  F  is  to  be  determined  by  considering  Fl  to  be  a  switch 
angle,  and  the  curve  C  F  to  commence  at  the  but-end  of  the  frog 
(§  50  or  §  51),  using  %g  instead  of  g  for  the  gauge. 

If  both  turnouts  turn  the  same  way,  as  in  figure  27,  the  third 
frog  J^a  is  on  a  turnout  A  FI  F*  from  the  inside  of  the  curve 
AF,  and  its  angle  and  position  may  be  determined  by  §  60 
or  §  72. 

Fig.  27. 


78.  Remarks.  1.  If  the  two  turnouts  of  figure  26  are  symmetri- 
cal and  tangent  to  the  straight  main  track,  the  chord  B  C  is  to 
the  chord  B  F  as  1  to  y2.  For  the  offset  from  the  tangent  B  F1 
to  C  is  \g,  and  the  offset  to  F  is  g,  and  these  tangent  offsets  or 
deflections  are  to  each  other  (§  18)  as  the  squares  of  the  chords 
B  C  and  B  F.  Therefore  B  C* :  B  F9  =  %g  :  g  =  1  :  2,  or  B  C  : 


I:f2;  whence  £  (7  =  ^  =  i  +/2  B  F  =  .707  5  JP,  nearly. 
2.  We  have  (§  66)  sin.  $  ^  =  ^=,  and  sin.  }  Jft  as  &  ss  ^-g-^. 

Jj  Jj  Jj  G         £  Jj  O 

Denote  the  whole  frog  angle  at  (7  by  F'  =  %Fi,  and  we  have 
sin.i^P'  =  X-^-TY-    Also,  since,  as  shown  above,  BF=BCv% 

&  -D   L> 

we   have  sin.  $  F  =  -=^-j= — -  .      Therefore,   sin.  i  F'  :  sin.  $  F  = 
-tf  L>  V '  &  ~ 

S-^TY  :  T.^    rt  =  V2  : 2,  or  sin.  ±F'  =  -^  sin.  i  J^=  .707  sin.  \  F, 
2BC    BC y%  2 

nearly. 


t)6  CIRCULAR   CURVES. 

3.  We  have  seen  (§§  66  and  71)  that  for  a  given  frog  angle  the 
length  of  the  chord  B  F  in  the  three  turnouts  represented  in 
figures  22,  23,  and  24  is  practically  the  same,  since  we  may  put  in 

the  three  cases  B  F  =    .     ,  „.    To  find  the  degree  of  each  of  the 
sin.  £  F 

three  turnout  curves,  we  have  only  to  find  the  central  angle  sub- 
tended by  a  chord  of  100  feet  (§  6).  Now,  in  the  three  cases  in 
question,  we  know  that  the  central  angles  B  E  F,  subtended  by 
the  equal  chords  B  F,  are,  respectively,  F,  F  +  K,  and  F—  K. 
The  central  angles  for  100  feet  chords  will  be  obtained  from  these 

100  100 

very  nearly  by  multiplying  by  JTJ,.    Denoting  the  fraction  --r, 


by  m  and  the  degrees  of  the  three  turnout  curves  by  AI,  A2,  and  A3, 
we  have  A!  =  m  F,  A2  =  m  (F  +  K),  A3  =  m  (F  -  K).  Now  m  K 
is  approximately  the  degree  of  the  curve  of  the  main  track  (figs. 
23  and  24)  since  K  is  the  central  angle  of  this  curve  for  a  chord 
approximately  equal  to  B  F.  Therefore,  denoting  the  degree  of 
the  main  track  by  A,  we  have,  approximately,  for  the  same  frog 
angle, 

Aa  =  AI  +  A,     A3  —  AI  —  A. 


Thus  in  the  example  of  §  71  (fig.  23),  where  n  =  8,  we  have  by 
Tab.  V.  the  degree  of  a  turnout  from  a  straight  line  AI  =  9°  31'. 
The  degree  of  the  main  track  is  here  A  =  3°.  Therefore  A2  = 
A!  +  A  =  12°  31',  the  degree  of  the  turnout  from  the  curve.  The 
radius  found  for  this  turnout  was  457.52  and  the  degree  corre- 
sponding would  be  12°  32'  53". 

It  appears,  then,  that  if,  for  a  given  frog,  we  take  from  Tab.  V. 
the  degree  AI  of  a  turnout  from  a  straight  main  track,  we  may 
obtain  approximately  the  degree  A2  of  a  turnout  from  the  inside 
of  a  curved  track  by  adding  to  AI  the  degree  of  the  main  track, 
and  the  degree  A3  of  a  turnout  from  the  outside  of  a  curved  track 
by  subtracting  from  AI  the  degree  of  the  main  track. 

ARTICLE  IV.  —  MISCELLANEOUS  PROBLEMS. 

79.  Problem.  Given  A  JB  =  a  (fig.  28)  and  the  perpendicular 
B  C  =  b,  to  find  the  radius  of  a  curve  that  shall  pass  through  G 
and  the  tangent  point  A. 

Solution.  Let  0  be  the  centre  of  the  curve,  and  draw  the  radii 
A  0  and  C  0  and  the  line  CD  parallel  to  A  B.  Then  in  the  right 


MISCELLANEOUS    PROBLEMS. 


67 


triangle   COT)  we  have   0  C2  =  CD'2  +  07)2.      But    0(7  =  72, 
CD  =  a,   and    0  D  =  A  0  -  A  D  =  R  -  b.      Therefore,    72*  = 

+  &2,  or  2  72  b  =  a2  +  62 ; 

*2 


Example.    Given  a  =  204  and  &  =  24,  to  find  72.    Here  R  = 


80.   Corollary  1.     If  R  and  &  are  given  to  find  AB  =  a, 
that  is,  to  determine  the  tangent  point  from  which  a  curve  of 


Fig.  28. 


given  radius  must  start  to  pass  through  a  given  point,  we  have 
(§  79)  2 Mb  =  a*  +  &2,  or  a2  =  2  JS  &  -  &2; 


Example.  Given  &  =  24  and  72  =  879,  to  find  a.  Here  a  = 
^24(1758-24)  =  V41616  =  204. 

81.  Corollary  2.  If  72  and  a  are  given,  and  b  is  required, 
we  have  (§79)  2  R  b  =  a*  +  &*,  or  &2  -  2  R  b  =  -  a*.  Solving 
this  equation,  we  find  for  the  value  of  b  here  required, 


82.   Problem.     Given  the  distance  A  C  =  c  (fig.  28)  and 
the  angle  B  A  C  =  A,  to  'find  the  radius  R  or  deflection  angle 


68  CIKCTJLAK   CURVES. 

D  of   a    curve,   that    shall  pass    through   C  and  the  tangent 
point  A. 

Solution.    Draw  0  E  perpendicular  to  A  C.    Then  the  angle 
AOE  =  iAOC  =  BAC=A(§2,  III.),  and  the  right  triangle 

^  Ogives  (Tab.  X.9MO  = 


..  . 

sin.  A 

50 
To  find  D,  we  have  (§  9)  sin.  D  =  -^  .    Substituting  for  72  its 

ic 
value  just  found,  we  have  sin.  Z>  =  50  -§-  -tP  —  -r  ; 


Example.    Given  c  =  285.4  and  A  —  5°,  to  find  R  and  D. 


Here  R  =  =  1637.3  ;  and  sin.  D  =  100  =  = 

sin.  5°  285.4        2.854 

sin.  1°  45'  or  D  =  1°  45'. 

83.  Problem.  Given  the  radius  R  or  the  deflection  angle 
D  of  a  curve,  and  the  angle  B  A  C  =  A  (fig.  %8},  made  by  any 
chord  with  the  tangent  at  A,  to  find  the  length  of  the  chord 
AC  =  c. 

1      x» 

Solution.    If  R  is  given,  we  have  (§  82)  R  =  —  —  -7  ; 

sin.  A. 

=  2R  sin.  A. 

10°  sin-  -^ 


T*  r>  •  /o  onx     •        :r> 

If  D  is  given,  we  have  (§  82)  sin.  Z)  = 

_  100  sin.  A 
sin.  Z> 

This  formula  is  useful  for  finding  the  length  of  chords,  when  a 
curve  is  laid  out  by  points  two,  three,  or  more  stations  apart. 
Thus,  suppose  that  the  curve  A  C  is  four  stations  long,  and  that 
we  wish  to  find  the  length  of  the  chord  A  C.  In  this  case  the 

angle  A  =  4  D  and  c  =  10°  sm'^  D  .    By  this  method  Table  II. 

sm.  D 
is  calculated. 

Example.    Given  R  =  2455.7,  or  D  =±  1°  10',  and  A  =  4°  40',  to 


MISCELLANEOUS   PROBLEMS. 


69 


find  c.    Here,  by  the  first  formula,  c  -  4911.4  sin.  4°  40'  =  399.59. 

100  sin.  4°  40' 
By  the  second  formula,  c  —  —  :  —  T^JTV  — 


84.  Problem.  Given  the  angle  of  intersection  KGB  —  I 
(fig.  %9\  and  the  distance  C  D  —  b  from  the  intersection  point  to 
the  curve  in  the  direction  of  the  centre,  to  find  the  tangent  AC  = 
T,  and  the  radius  A  0  =  R. 


Fig.  29. 


Solution.  In  the  triangle  A  D  C  we  have  sin.  C  A  D  :  sin. 
AD  C-  CD\  AC.  But  CAD  =  ^AOD  =  lI  (§2,  III.  and 
VI.),  and  as  the  sine  of  an  angle  is  the  same  as  the  sine  of  its  sup- 
plement, sin.  ADC—  sin.  ADE  =  cos.  D  A  E  =  cos.  £  /.  More- 
over, C  D  —  b  and  A  C  =  T.  Substituting  these  values  in  the 
preceding  proportion,  we  have  sin.  \  / :  cos.  £  I  =  b  :  T,  or  T  = 

6?°S'*J;  whence  (Tab.  X.  33) 


•  T  =  b  cot.  i  7. 

To  find  R,  we  have  (§  5)  R  =  Tcoi.  i  I.    Substituting  for  T  its 
value  just  found,  we  have 

R  =  b  cot.  i  /cot.  i  I. 


70  CIKCULAK   CURVES. 

Example.    Given  /  =  30°,  b  —  130,  to  find  2*  and  R.    Here 

b  =  130  2.113943 

i  7  =  7°  30'       cot.  0.880571 

T=  987.45  2.994514 

i/=15°  cot.  0.571948 

R  =  3685.21  3.566462 

85.  Problem.  Given  the  angle  of  intersection  KG  B  —  I 
(fig.  29\  and  the  tangent  A(J—T,  or  the  radius  A  0  =  R,  to  find 
CD=b. 

Solution.    If  T  is  given,  we  have  (§  84)  T  =  I  cot.  £  /,  or  b  = 

T 
cot.i/' 

B^~  .-.&=  Ttan.iZ 

If  .72  is  given,  we  have  (§  84)  R  =  b  cot.  J  7  cot.  •£  /,  or  &  = 


.  •  .  b  =  R  tan.  i  /tan.  -J  Z 


Example.  Given  J=  27°,  T=  600  or  j^  =  2499.18,  to  find  b. 
Here  &  =  600  tan.  6°  45'  =  71.01,  or  b  =  2499.18  tan.  6°  45'  tan. 
13°  30'  =  71.01. 

The  distance  b  from  the  intersection  point  to  the  curve  in  the 
direction  of  the  centre  is  usually  called  the  external,  and  this  term 
is  adopted  in  Table  III. 

86.  Problem.  Given  the  angle  of  intersection  I  of  two  tan- 
gents A  C  and  B  C  (fig.  30\  to  find  the  tangent  point  A  of  a  curve 
that  shall  pass  through  a  point  E,  given  by  C  D  =  a,  D  E  =•  b, 
and  the  angle  C  D  E  =  -J  /. 

Solution.  Produce  D  E  to  the  curve  at  G,  and  draw  C  0  to  the 
centre  0.  Denote  D  F  by  c.  Then  in  the  right  triangle  CDF 
we  have  (Tab.  X.  11)  D  F  =  C  D  cos.  CD  F,  or 

jgjp"  c  —  a  cos.  •$•  /. 

Denote  the  distance  A  D  from  D  to  the  tangent  point  by  xt 
Then,  by  Geometry,  a2  =  D  E  x  D  O.  But  D  0  -  D  F  +  F  Q  = 


D F  + 

b),  and 


MISCELLANEOUS    PROBLEMS.  71 

=  2  c  —  b.    Therefore,  x9  =  b  (2  c  — 


Having  thus  found  A  D,  we  have  the  tangent  A  C  =  A  D  + 
D  C  =  x  +  a.  Hence,  R  or  D  may  be  found  (§  5  or  §  11). 

If  the  point  E  is  given  by  E  H  and  C  H  perpendicular  to  each 
other,  a  and  b  may  be  found  from  these  lines.  For  a  =  C  H  + 

DH=CH+  EHcoi.  \  /(Tab.  X.  9),  and  b  =  D  E  = 


Example.  Given  /=  20°  16',  a  =  600,  and  b  =  80,  to  find  x 
and  R.  Here  c  =  600  cos.  10°  8'  =  590.64,  2  c  —  b  =  1101.28,  and 
x  =  \/80  x  1101.28  =  296.82.  Then  T  —  600  +  296.82  =  896.82, 
and  R  =  896.82  cot.  10°  8'  =  5017.82. 


87.  Problem.  Given  the  tangent  A  C  (fig.  31),  and  the 
chord  A  B,  uniting  the  tangent  points  A  and  B,  to  find  the  radius 
AO  =  R. 

Solution.  Measure  or  calculate  the  perpendicular  C  D.  Then 
if  CD  be  produced  to  the  centre  0,  the  right  triangles  ADC  and 


72  CIRCULAR   CURVES. 

C  A  0,  having  the  angle  at  C  common,  are  similar,  and  give  CD  \ 
AD  =  AC:AO,OT 

_AD x  A C 
CD       ' 

If  it  is  inconvenient  to  measure  the  chord  AB,  a  line  E F^ 
parallel  to  it,  may  be  obtained  by  laying  off  from  C  equal  dis- 
tances CE  and  C  F.    Then  measuring  EG  and  #  (7,  we  have, 
from  the  similar  triangles  E  G  C  and  CAO,  CG  :  GE -  AC : 
GE  xAC 


Example.    Given  A  C  —  246  and  A  D  =  240,  to  find  R.    Here 

CD  =  54,  and  R  =  24°*.246  =  1093.33. 
54 


88.  Problem.  Given  the  radius  AO  =  R  (fig.  31\  to  find 
the  tangent  AC  —  T  of  a  curve  to  unite  two  straight  lines  given 
on  the  ground. 


Pig.  31. 


Solution.  Lay  off  from  the  intersection  C  of  the  given  straight 
lines  any  equal  distances  C  E  and  C  F.  Draw  the  perpendicular 
C  G  to  the  middle  of  E  F,  and  measure  GE  and  C  G.  Then  the 


MISCELLANEOUS    PROBLEMS.  73 

right  triangles  E  Or  C  and  C  A  0,  having  the  angle  at  C  common, 
are  similar,  and  give  G  E  :  C  G  =  A  0  :  A  C,  or 

CGxAO 
W  GE       • 

By  this  problem  and  the  preceding  one,  the  radius  or  tangent 
points  of  a  curve  may  be  found  without  an  instrument  for  measur- 
ing angles. 

Example.  Given  R  =  1093£,  G  E  =  80,  and  C  G  -  18,  to  find 
K  Here  T  =**£**  =  246. 

89.  Problem.  To  find  the  angle  of  intersection  I  of  two 
straight  lines,  when  the  point  of  intersection  is  inaccessible,  and  to 
determine  the  tangent  points,  when  the  length  of  the  tangents  is 
given. 

Solution.  I.  To  find  the  angle  of  intersection  /.  Let  A  C  and 
C  V  (fig.  32)  be  the  given  lines.  Sight  from  some  point  A  on  one 
line  to  a  point  B  on  the  other,  and  measure  the  angles  CAB  and 
T B  V.  These  angles  make  up  the  change  of  direction  in  passing 
from  one  tangent  to  the  other.  But  the  angle  of  intersection 
(g  2)  shows  the  change  of  direction  between  two  tangents,  and  it 
must,  therefore,  be  equal  to  the  sum  of  C  A  B  and  T  B  V,  that  is, 

I=CAB  +  TB  V. 


But  if  obstacles  of  any  kind  render  it  necessary  to  pass  from 
A  C  to  B  V  by  a  broken  line,  as  A  D  E  F  B,  measure  the  angles 
CAD,  ND  E,  PEF,  RFB,  and  SB  V,  observing  to  note  those 
angles  as  minus  which  are  laid  off  contrary  to  the  general  direc- 
tion of  these  angles.  Thus  the  general  direction  of  the  angles  in 
this  case  is  to  the  right ;  but  the  angle  PEF  lies  to  the  left  of 
D  E  produced,  and  is  therefore  to  be  marked  minus.  The  angles 
to  be  measured  show  the  successive  changes  of  direction  in  passing 
from  one  tangent  to  the  other.  Thus  CAD  shows  the  change  of 
direction  between  the  first  tangent  and  A  D,  ND  E  shows  the 
change  between  AD  produced  and  DE,  PEF  the  change  be- 
tween DE  produced  and  E  F,  RFB  the  change  between  E  F 
produced  and  F  B,  and,  lastly,  SB  V  the  change  between  B  F 
produced  and  the  second  tangent.  But  the  angle  of  intersection 
(§  2)  shows  the  change  of  direction  in  passing  from  one  tangent  to 


74  CIRCULAR   CURVES. 

another,  and  it  must,  therefore,  be  equal  to  the  sum  of  the  partial 
changes  measured,  that  is, 

NDE-PEF+RFB  +  SBV. 


II.  To  determine  the  tangent  points.  This  will  be  done  if  we 
find  the  distances  A  C  and  B  C ;  f or  then  any  other  distances  from 
C  may  be  found.  It  is  supposed  that  the  distance  A  B,  or  the 
distances  A  Z>,  D  E,  E  F,  and  F  B  have  been  measured. 

If  one  line  A  B  connects  A  and  B,find  A  C  and  B  C  in  the  tri- 
angle ABC.  For  this  purpose  we  have  one  side  A  B  and  all  the 


If  a  broken  line  ADEFB  connects  A  and  B,  let  fall  a  per- 
pendicular B  G  from  B  upon  A  C,  produced  if  necessary,  and 
find  A  G  and  B  G  by  the  usual  method  of  working  a  traverse. 
Thus,  if  A  C  is  taken  as  a  meridian  line,  and  D  K,  EL,  and  F M 
are  drawn  parallel  to  A  C,  and  D  H,  E K,  and  F L  are  drawn 
parallel  to  B  G,  the  difference  of  latitude  A  G  is  equal  to  the  sum 
of  the  partial  differences  of  latitude  AH,DK,EL,  and  F  M, 
and  the  departure  B  G  is  equal  to  the  sum  of  the  partial  depart- 
ures D  H,  EK,  FL,  and  B  M.  To  find  these  partial  differences 
of  latitude  and  departures,  we  have  the  distances  AD,DE,EF, 
and  F  B,  and  the  bearings  may  be  obtained  from  the  angles  al- 
ready measured.  Thus  the  bearing  of  A  D  is  CAD,  the  bearing 
of  DE  is  KDE  =  KDN  +  NDE  =  CAD  +  NDE,  the  bear- 
ing of  EFi$LEF=LEP—  PEF=KDE— PEf,  and 


MISCELLANEOUS   PROBLEMS. 


75 


the  bearing  of  F  B  is  M  F  B  =  M  F  R  +  R  F  B  =  L  E  F  + 
RFB;  that  is,  the  bearing  of  each  line  is  equal  to  the  algebraic 
sum  of  the  preceding  bearing  and  its  own  change  of  direction. 
The  differences  of  latitude  and  the  departures  may  now  be  ob- 
tained from  a  traverse  table,  or  more  correctly  by  the  formulas  : 
Diff  .  of  lat.  =  dist.  x  cos.  of  bearing  ;  dep.  =  dist.  x  sin.  of  bearing. 
Thus,  A  H  =  A  D  cos.  CA  D,  and  DH  =  A  D  sin.  CAD. 
Having  found  A  G  and  B  G,  we  have,  in  the  right  triangle 

7?  C1 

EGG  (Tab.  X.  9),  G  C  =  B  G  cot,  BCG,an&BC  =  -  —  ^7775  . 

sin.  jj  (j  (j 

But  BCG  =  180°  -  7.     Therefore,   cot.  BCG=-  cot.  /,  and 
sin.  B  C  G  =  sin.  I.    Hence  G  C  =  -  B  G  cot.  /,  and  B  C  = 

Then,  since  AC=  AG  +  G  C,  we  have 


~T. 

sin.  / 


When  /  is  between  90°  and  180°,  as  in  the  figure,  cot.  /  is  nega- 
tive, and  —  B  G  cot.  /  is,  therefore,  positive.  When  /  is  less  than 
90°,  G  will  fall  on  the  other  side  of  C ;  but  the  same  formula  for 
A  C  will  still  apply ;  for  cot.  /  is  now  positive,  and  consequently, 
—  B  G  cot.  /  is  negative,  as  it  should  be,  since,  in  this  case,  A  C 
would  equal  A  G  minus  G  C. 

Example.  Given  A  D  =  1200,  D  E  =  350,  E  F  =  300,  FB  = 
310,  CAD  =  2Q°,  ND  #  =  44°,  PE  F=  -  25°;  R  FB  =  Z\\ 
and  SB  V=  30°,  to  find  the  angle  of  intersection  /,  and  the  dis- 
tances A  C  and  B  C. 

Here  J=  20°  +  44°  -  25°  +  31°  +  30°  =  100C.  To  find  A  G 
and  B  G,  the  work  may  be  arranged  as  in  the  following  table : — 


Angles  to 
the  Right. 

Bearings. 

Distances. 

N. 

E. 

20 
44 
—25 
31 

N.  20  E. 
64 
39 
70 

1200 
350 
300 
310 

1127.63 
153.43 
233.14 
106.03 

410.42 
314.58 

188.80 
291.30 

1620.23 

1205.10 

The  first  column  contains  the  observed  angles.    The  second  con- 
tains the  bearings,  which  are  found  from  the  angles  of  the  first 


76 


CIRCULAR   CURVES. 


column,  in  the  manner  already  explained.  A  C  is  considered  as 
running  north  from  A,  and  the  bearings  are,  therefore,  marked 
N.  E.  The  other  columns  require  no  explanation.  We  find 
A  a  =  1620.23  and  B  G  =  1205.10.  Then  GC=-BG  cot.  1 
=  —  1205.1  x  cot.  100°  =  212.49.  This  value  is  positive,  because 
it  is  the  product  of  two  negative  factors,  cot.  100°  being  the  same 
as  —  cot.  80°,  a  negative  quantity.  Then  A  C  =  A  G  +  G  C  — 


1620.23  +  212.49  =  1832.72,  and  B  C  = 


=  1223.69.    Hav- 


0 
sm.  100 

ing  thus  found  the  distances  of  A  and  B  from  the  point  of  inter- 
section, we  can  easily  fix  the  tangent  points  for  tangents  of  any 
given  length. 

90.  Problem.     To  lay  out  a  curve,  when  an  obstruction  of 
any  kind  prevents  the  use  of  the  ordinary  methods. 


Fig.  33, 


Solution.  First  Method.  Suppose  the  instrument  to  be  placed 
at  A  (fig.  33),  and  that  a  house,  for  instance,  covers  the  station  at 
B,  and  also  obstructs  the  view  from  A  to  the  stations  at  D  and  E. 
Lay  off  from  A  (7,  the  tangent  at  A,  such  a  multiple  of  the  deflec- 
tion angle  D,  as  will  be  sufficient  to  make  the  sight  clear  the  ob- 
struction. In  the  figure  it  is  supposed  that  4  D  is  the  proper  an- 


MISCELLANEOUS   PROBLEMS.  77 

gle.  The  sight  will  then  pass  through  F,  the  fourth  station  from 
A,  and  this  station  will  be  determined  by  measuring  from  A  the 
length  of  the  chord  A  F,  found  by  §  83  or  by  Table  II.  From  the 
station  at  F  the  .stations  at  D  and  E  may  afterwards  be  fixed,  by 
laying  off  the  proper  deflections  from  the  tangent  at  F. 

Second  Method.  This  consists  in  running  an  auxiliary  curve 
parallel  to  the  true  curve,  either  inside  or  outside  of  it.  For  this 
purpose  lay  off  perpendicular  to  A  C,  the  tangent  at  A,  a  line 
A  A  of  any  convenient  length,  and  from  A  a  line  A  C'  parallel 
to  A  C.  Then  AC'  is  the  tangent  from  which  the  auxiliary  curve 
A  E'  is  to  be  laid  off.  The  stations  on  this  curve  are  made  to 
correspond  to  stations  of  100  feet  on  the  true  curve,  that  is,  a  ra- 
dius through  B'  passes  through  B,  a  radius  through  D'  passes 
through  D,  &c.  The  chord  A  B'  is,  therefore,  parallel  to  AB, 
and  the  angle  C'  A  B'  —  C  A  B  ;  that  is,  the  deflection  angle  of 
the  auxiliary  curve  is  equal  to  that  of  the  true  curve.  It  remains 
to  find  the  length  of  the  auxiliary  chords  A  B',  B'  D',  &c.  Call 
the  distance  A  A'  =  b.  Then  the  similar  triangles  ABO  and 
A'  B  '  0  give  A  0  :  A'  0  =  A  B  :  A'  B  ',  or  R  :  R  -  b  =  100  :  A'B  '. 


Therefore,  A  B'  =  ~      =  ioo  -~.    If  the  auxiliary 

£1  jft 

curve  were  on  the  outside  of  the  true  curve,  we  should  find  in  the 
same  way  A  B  '  =  100  +  p  .  It  is  well  to  make  b  an  aliquot 

part  of  R-,  for  the  auxiliary  chord  is  then  more  easily  found. 

7~> 

Thus,  if  n  is  any  whole  number,  and  we  make  b  =  —  ,  we  have 

71 

A'B'  =  m±  ^  =  100  ±  —  .    If,  for  example,  b  =  ^~,  we 
jK  fl  lUu 

have  n  —  100,  and  A  B  =  100  ±  1  =  101  or  99.  When  the  aux- 
iliary curve  has  been  run,  the  corresponding  stations  on  the  true 
curve  are  found,  by  laying  off  in  the  proper  direction  the  distances 
B  B',  D  D',  &c.,  each  equal  to  b. 

91.  Problem.  Having  run  a  curve  A  B  (fig.  34\  to  change 
the  tangent  point  from  A  to  (7,  in  such  a  way  that  a  curve  of  the 
same  radius  may  strike  a  given  point  D. 

Solution.  Measure  the  distance  B  D  from  the  curve  to  D  in  a 
direction  parallel  to  the  tangent  C  E.  This  direction  may  be 
sometimes  judged  of  by  the  eye,  or  found  by  the  compass.  A  still 
more  accurate  way  is  to  make  the  angle  D  B  E  equal  to  the  inter- 


78 


CIRCULAR    CURVES. 


section  angle  at  E,  or  to  twice  B  A  E,  the  total  deflection  angle 
from  A  to  B ;  or  if  A  can  be  seen  from  B,  the  angle  DBA  may 
be  made  equal  to  B  A  E. 

Measure  on  the  tangent  (backward  or  forward,  as  the  case  may 
be)  a  distance  A  C  =  B  Z),  and  C  will  be  the  new  tangent  point  re- 
quired. For,  if  CHbv  drawn  equal  and  parallel  to  A  F,  we  have 
F  H  equal  and  parallel  to  A  C,  and  therefore  equal  and  parallel 
to  B  D.  Hence  DH=£F  =  AF=CH,  and  D  H  being  equal 
to  Off,  &  curve  of  radius  C  H  from  the  tangent  point  C  must  pass 
through  D. 


92.  Problem.  Having  run  a  curve  A  B  (fig.  55)  of  radius 
R  or  deflection  angle  D,  terminating  in  a  tangent  B  D,  to  find  the 
radius  R '  or  deflection  angle  D'  of  a  curve  A  C,  that  shall  termi- 
nate in  a  given  parallel  tangent  C  E. 

Solution.  Since  the  radii  B  F  and  C  G  are  perpendicular  to 
the  parallel  tangents  C  E  and  B  D,  they  are  parallel,  and  the  an- 
gle A  G  C-  A  FB.  Therefore,  A  C G,  the  half-supplement  of 
A  G  C,  is  equal  to  A  B  F,  the  half -supplement  of  A  F  B.  Hence 
A  B  and  B  C  are  in  the  same  straight  line,  and  the  new  tangent 
point  C  is  the  intersection  of  A  B  produced  with  C  E. 

Represent  A  B  by  c,  and  A  C  =  c  +  B  C  by  c'.  Measure  B  (7, 
or,  if  more  convenient,  measure  D  C  and  find  B  C  by  calculation. 

DC 


To  calculate  B  C  from  D  C,  we  have  B  C  = 


1  (Tab.  X.  9) 


sin.  DB  C  v 
and  the  angle  DBG-  ABK—BAK,  the  total  deflection  from 


MISCELLANEOUS   PKOBLEMS. 


79 


A  to  B.    Then  the  triangles  A  FB  and  A  G  C  give  A  B  :  A  C  • 
BF :  CG,  or  c  :  c'  =  R  :  R' ', 


To  find  D,  we  have (§  10)  R '  =  -rr, ,  and  R  =  -= .    Sub- 
sin.  D  '  sm.D 

stituting  these  values  in  the  equation  for  R ',  we  have  - — —  = 

c'         50  Sm' 

-  x  — =~; 

c       sin.  D 

.  • .  sin.  D'  •=  —  sin.D. 

c 


93.  Problem.  Given  the  length  of  two  equal  chords  A  C  and 
B  C  (fig.  36),  and  the  perpendicular  CD,  to  find  the  radius  R  of 
the  curve. 

Solution.  From  0,  the  centre  of  the  curve,  draw  the  perpen- 
dicular 0 E.  Then  the  similar  triangles  0 B E  and  BCD  give 
BO:BE=BC:  CD,  or  R  :  \  B  C  =  B  C  :  CD.  Hence 

BC* 


R  = 


2CD' 


This  problem  serves  to  find  the  radius  of  a  curve  on  a  track 
already  laid.  For  if  from  any  point  C  on  the  curve  we  measure 
two  equal  chords  A  C  and  B  C,  and  also  the  perpendicular  CD 


80 


CIRCULAR   CURVES. 


from  C  upon  the  whole  chord  AB,  we  have  the  data  of  this 
problem. 

94.  Problem.     To  draw  a  tangent  F  Gr  (fig.  36)  to  a  given 
curve  from  a  given  point  F. 


Fig.  36. 


Solution.  On  any  straight  line  FA,  which  cuts  the  curve  in 
two  points,  measure  FG  and  FA,  the  distances  to  the  curve. 
Then,  by  Geometry, 

J^~  FG  =  ^FC  x  FA. 

This  length  being  measured  from  F,  will  give  the  point  G. 
When  F  G  exceeds  the  length  of  the  chain,  the  direction  in  which 
to  measure  it,  so  that  it  will  just  touch  the  curve,  may  be  found 
by  one  or  two  trials. 

95.  Problem.  Having  found  the  radius  A  0  —  R  of  a  curve 
(fig.  31\  to  substitute  for  it  two  radii  A  E  =  Ri  and  D  F  =  R* , 
the  longer  of  which  A  E  or  BE'  is  to  be  used  for  a  certain  dis- 
tance only  at  each  end  of  the  curve. 

Solution.  Assum.e  the  longer  radius  of  any  length  which  may 
he  thought  proper,  and  find  (g  9)  the  corresponding  deflection 
angle  DI  .  Suppose  that  each  of  the  curves  A  D  and  B  D '  is  100 
feet  long.  Then  drawing  C  0,  we  have,  in  the  triangle  FOE, 
OE  :FE=sm.  0  F  E  :  sin.  FOE.  But  the  side  OE  =  AE  — 

Rl  -  R*,  the  angle  .F  0  i7  = 


MISCELLANEOUS   PROBLEMS. 


81 


180°  -  A  0  C  =  180°  -  i  /,  and  the  angle   OFE  =  AOF- 
=  iI-2Dl,  since   0  E  F  =  2  Dl   (§  7).    Substituting 


Fig.  37. 


these  values,  and  recollecting  that  sin.  (180°  —  -J-  /)  =  sin.  £  /,  we 
have  Ei  —  R  :  Hi  —  R*  —  sin.  (|  /  —  2  DI)  :  sin.  -£  /.    Hence 


HP 


sin.  (|  J- 2  A) 


7?  2  is  then  easily  found,  and  this  will  be  the  radius  from  D  to  Z>', 
or  until  the  central  angle  D  F  D '  —  I  —  4  Z>i. 

The  object  of  this  problem  is  to  furnish  a  method  of  flattening 
the  extremities  of  a  sharp  curve.  It  is  not  necessary  that  the  first 
curve  should  be  just  100  feet  long;  in  a  long  curve  it  may  be 
longer,  and  in  a  short  curve  shorter.  The  value  of  the  angle  at 
E  will  of  course  change  with  the  length  of  A  Z>,  and  this  angle 
must  take  the  place  of  2Di  in  the  formula.  The  longer  the  first 
curve  is  made,  the  shorter  the  second  radius  will  be.  It  must  also 
be  borne  in  mind,  in  choosing  the  first  radius,  that  the  longer  the 
first  radius  is  taken,  the  shorter  will  be  the  second  radius. 
7 


82 


CIRCULAR   CURVES. 


Example.  Given  R  =  1146.28  and  I—  45°,  to  find  R*,  if  Rl  is 
assumed  =  1910.08,  and  A  D  and  B  D '  each  100.  Here,  by  Table 
I.,A  =  1°30'.  Then 


R^  -  R  =  763.8 
i/=22°  30' 

|/-.  2  A  =  19°  30' 
Ri-R*  =  875.64 

2.882980 
sin.  9.582840 

2.465820 
sin.  9.523495 

2.942325 

.-.R^Rt-  875.64  =  1034.44 

96.  Problem.  To  locate  the  second  branch  of  a  compound 
or  reversed  curve  from  a  station  on  the  first  branch. 

Solution.  Let  A  B  (fig.  38)  be  the  first  branch  of  a  compound 
curve,  and  D  its  deflection  angle,  and  let  it  be  required  to  locate 
the  second  branch  A  B',  whose  deflection  angle  is  D',  from  some 
station  B  on  A  B. 

Let  n  be  the  number  of  stations  from  A  to  B,  and  n'  the  number 
of  stations  from  A  to  any  station  B '  on  the  second  branch.  Rep- 


_T 


Fig.  38. 


resent  by  V  the  angle  ABB',  which  it  is  necessary  to  lay  off  from 
the  chord  B  A  to  strike  B'.  Let  the  corresponding  angle  A  B'  B 
on  the  other  curve  be  represented  by  V.  Then  we  have  V  + 
V  —  180°  —  BAB'.  But  if  T T'  be  the  common  tangent  at  A, 
we  have  T  A  B  +  T'  A  B'  =  n  D  +  n'  D'  =  180°  —  BAB'. 
Therefore,  V  +  V  =  n  D  +  n'  D' .  Next  in  the  triangle  ABB' 
we  have  sin.  V  :  sin.  V  =  A  B  :  A  B '.  But  A  B  :  A  B '  —  n  :  n', 
nearly,  and  sin.  V  :  sin.  V=  V  :  V,  nearly.  Therefore  we  have 

approximately  V  :V=n  :  n',  or  V  —  —,  V.  Substituting  this 
value  of  V  in  the  equation  for  V  +  V,  we  have  V  +  —,V  = 


MISCELLANEOUS   PROBLEMS.  83 

n  D  -f-  n'  D  '.    Therefore,  n'V+  n  V  =  ri  (n  D  +  ri  D  '),  or 

n'(nD  +  n'D') 
n  +  ri 

The  same  reasoning  will  apply  to  reversed  curves,  the  only 
change  being  that  in  this  case  V  +  V  =  nD  —  n'  D',  and  conse- 
quently 

ri(nD-riD') 

uHy?  ,  • 

n  +  n 

When  in  this  last  formula  n'  D'  becomes  greater  than  nD,  V  be- 
comes minus,  which  signifies  that  the  angle  V  is  to  be  laid  off 
above  B  A  instead  of  below. 

This  problem  is  particularly  useful,  when  the  tangent  point  of 
a  curve  is  so  situated,  that  the  instrument  cannot  be  set  over  it. 
The  same  method  is  applicable,  when  the  curve  A  B'  starts  from 
a  straight  line  ;  for  then  we  may  consider  A  B  '  as  the  second 
branch  of  a  compound  curve,  of  which  the  straight  line  is  the 
first  branch,  having  its  radius  equal  to  infinity,  and  its  deflection 
angle  D  =  0.  Making  D  =  0,  the  formula  for  V  becomes 


_ 

n  +  n'  ' 

When  n  and  n'  are  each  1,  the  formula  for  V  is  in  all  cases  ex- 
act; for  then  the  supposition  that  V  :  V=  n  :  n'  is  strictly  true, 
since  A  B  will  equal  A  B',  and  V  and  V,  being  angles  at  the 
base  of  an  isosceles  triangle,  will  also  be  equal.  Making  n  and  n' 
equal  to  1,  we  have 


When  the  curve  starts  from  a  straight  line,  this  formula  becomes, 
by  making  D  =  0, 

F=*Z>'. 

We  have  seen  that  when  n  or  n'  is  more  than  1,  the  value  of  V 
is  only  approximate.  It  is,  however,  so  near  the  truth,  that  when 
neither  n  nor  n'  exceeds  3,  the  error  in  curves  up  to  5°  or  6°  varies 
from  a  fraction  of  a  second  to  less  than  half  a  minute.  The  exact 
value  of  V  might  of  course  be  obtained  by  solving  the  triangle 
A  B  B',  in  which  the  sides  AB  and  AB'  may  be  found  from 
Table  II.,  and  the  included  angle  at  A  is  known.  The  extent  to 
which  these  formulae  may  be  safely  used  may  be  seen  by  the  fol- 
lowing table,  which  gives  the  approximate  values  of  Ffor  several 


84 


CIRCULAR   CURVES. 


different  values  of  n,  n\  D,  and  Z>',  and  also  the  error  in  each 
case: 


Compound  Curves. 

Reversed  Curves. 

n. 

D. 

n'. 

D'. 

F. 

Error. 

n. 

D. 

n'. 

D'. 

V. 

Error. 

0 

0 

0        / 

// 

o 

0 

0        / 

„ 

1 

0 

5 

1 

410 

0.9 

1 

3 

4 

3 

712 

27.2 

1 

0 

5 

3 

1230 

25.3 

2 

3 

4 

3 

4    0 

23.5 

2 

0 

8 

3 

524 

22.1 

3 

3 

4 

3 

1  42f 

8.3 

3 

0 

3 

3 

430 

29.7 

3 

i 

3 

3 

345 

24.0 

1 

1 

5 

3 

1320 

18.6 

2 

1 

1 

4 

040 

0.1 

2 

f 

1 

3 

1  20 

0.7 

2 

1 

4 

2 

4    0 

11.0 

2 

2 

3 

3 

748 

15.0 

1 

6 

2 

6 

4    0 

23.5 

2 

2 

4 

3 

1040 

24.7 

1 

5 

3 

5 

730 

51.8 

3 

3 

3 

4 

1030 

54.0 

2 

3 

5 

3 

625f 

52.8 

As  the  given  quantities  are  here  arranged,  the  approximate 
values  of  V  are  all  too  great ;  but  if  the  columns  n  and  n'  and 
the  columns  D  and  D'  were  interchanged^  and  F  calculated,  the 
approximate  values  of  V  would  be  just  as  much  too  small,  the 
column  of  errors  remaining  the  same. 


97.   Problem.     To  measure  the  distance  across  a  river  on  a 
given  straight  line. 


Fig.  39. 


Solution.  First  Method.  Let  A  B  (fig.  39)  [be  the  required 
distance.  Measure  a  line  A  C  along  the  bank,  and  take  the  an- 
gles BAG  and  A  C  B.  Then  in  the  triangle  A  B  C  we  have  one 
side  and  two -angles  to  find  A  B. 

If  A  C  is  of  such  a  length  that  an  angle  AC B  —  ^DAC  can 


MISCELLANEOUS    PROBLEMS. 


85 


be  laid  off  to  a  point   on  the  farther  side,  we  have  A  S  C  = 
=  ACB.     Therefore,  without  calculation,  A  B  =  A  C. 


Fig.  40. 


Second  Method.  Lay  off  A  C  (fig.  40)  perpendicular  to  A  B. 
Measure  A  C,  and  at  C  lay  off  C  D  perpendicular  to  the  direction 
C  B,  and  meeting  the  line  of  A  B  in  D.  Measure  A  D.  Then  the 
triangles  A  C  D  and  A  B  G  are  similar,  and  give  A  D  :  A  C  = 

A  r* 

AC'.AB.    Therefore,  AB  =  ^^. 

If  from  (7,  determined  as  before,  the  angle  A  C  B'  be  laid  off 
equal  to  A  C  B,  we  have,  without  calculation,  A  B  =  A  B'. 

Third  Method.  Measure  a  line  A  D  (fig.  41)  in  an  oblique  di- 
rection from  the  bank,  and  fix  its  middle  point  C.  From  any 


Fig.  41. 


convenient  point  E  in  the  line  of  A  B,  measure  the  distance  E  (7, 
and  produce  EC  until  C F =  E  C.    Then,  since  the  triangles 


86 


CIRCULAR   CURVES. 


ACE  and  D  C  F  are  similar  by  construction,  we  see  that  D  F  is 
parallel  to  E  B.  Find  now  a  point  6r,  that  shall  be  at  the  same 
time  in  the  line  of  C  B  and  of  D  I1,  and  measure  O  D.  Then  the 
triangles  ABC  and  D  &  C  are  equal,  and  O  D  is  equal  to  the  re- 
quired distance  A  B. 

As  the  object  of  drawing  E  F  is  to  obtain  a  line  parallel  to  A  B, 
this  line  may  be  dispensed  with,  if  by  any  other  means  a  line  O  F 
be  drawn  through  D  parallel  to  A  B.  A  point  G  being  found  on 
this  parallel  in  the  line  of  C  J5,  we  have,  as  before,  G  D  =  AB. 

98.  Problem,  To  change  a  tangent  point  so  that  the  tan- 
gent may  pass  through  a  given  point. 

Solution.  If  the  given  point  is  at  a  considerable  distance  but 
visible,  let  C  (fig.  42)  be  the  distant  point  and  D  the  required  tan- 


Fig.  42. 


gent  point.  Estimate  the  probable  position  of  Z>,  and  at  A,  a 
station  back  of  D  but  near  to  it,  measure  the  angle  B  A  C  made 
by  A  C  with  the  tangent  at  A.  Then,  as  the  angle  at  C  is  sup- 
posed to  be  very  small,  the  chord  A  E  will  be  nearly  parallel  to 
D  (7,  and  D  may  be  taken  to  be  midway  between  A  and  E.  The 
angle  B  A  D,  which  fixes  the  position  of  Z),  will  therefore  equal 
i  B  A  <?,  very  nearly.  Or,  by  §  83,  compute  AE=ZR  sin.  B  A  C, 
and  we  shall  have  the  chord  A  D  =  $  A  E,  very  nearly.  If  the 
distance  A  C  is  not  very  great,  A  C  and  E  C  may  be  measured. 
Then  (§  94)  D  C  =  \/A  C  x  E  C. 

If  the  point  C  is  given  by  A  B  —  a  (fig.  43  or  44)  measured  on 
a  tangent  at  A,  and  B  C  =  ~b  at  right  angles  to  A  B,  draw  C  E 


MISCELLANEOUS    PROBLEMS. 


87 


parallel  to  A  B  to  meet  0  A,  produced  if  necessary.    Then,  in  the 
first  case  (fig.  43),  we  have  the  required  angle  A  OD  —  A  0  C  — 


Fig.  44. 


DOC. 


R 


Hence,  the  required  angle  is  determined. 


In  the  second  case  (fig.  44)  we  have  the  required  angle  AOD  = 

DOC  —  AOC.    But  cos.  DOC= =      .  2  =  ,  and 

tan.  A  0  C  =  -=-^  —  —  — - .     Hence,  the  required  angle  A  0  D  is 

,,  .  T  Jit     (J  /**     -L     /I 

determined. 


, . 

0 


99.  Problem.     To  connect  two  curves  by  a  common  tangent. 

Solution.  When  both  curves  turn  the  same  way  (fig.  45),  run  a 
line  A  B  cutting  both  curves  in  such  a  way  as  to  make  the  middle 
ordinates  E  G  and  F H  as  nearly  equal  as  can  conveniently  be 


Fig.  45. 


CIRCULAR    CURVES. 


done.  Measure  A  B  —  a  and  the  tangential  angles  C  A  B  —  A 
and  D  B  A  —  B.  Let  E'  F'  be  the  required  common  tangent, 
and  draw  0  E  and  P  F  perpendicular  to  A  B,  and  F'  K  parallel 
to  A  B.  Let  A  0  —  R  and  B  P  =  R '.  Then  the  required  angle 
CAE'^^AOE'  —  ^A  +  %EOE'  =  ^A  -f  %E'  F'  K.  Now 

EG  —  FH 
tan.  E'  F'  K  =  —  „--= ,  nearly  — 


Hence  C  A  E'  is  determined. 


a  —  It  sin.  A  —  R  '  sin.  B 
We  have  also  the  angle  P  B  F1  — 


When  the  curves  turn  opposite  ways  (fig.  46),  A  H  =  a  should 
be  run  outside  the  second  curve,  making  F  H  as  nearly  equal 


Fig.  46. 


to  E  G  as  can  conveniently  be  done.    FH  must  be  measured. 

Then  the  required  angle  CA  E'  =  %A  OE1  =  \A  +  $EOE'  = 

~&  r< ~p  TT 

4-  A  +  \E'  F'  K.    Now  tan.  E '  F '  K  —  —     •=-= ,  nearly  = 

a  t>    •    i  Tjy  TT  G  H 

'—^i — : ; .     Hence  C  A  E'  is  determined. 

a  —  R  sin.  A 

In  both  these  cases  E  G  has  been  supposed  larger  than  .F  77. 
If  E  G  is  smaller  than  F  H,  the  point  E'  will  fall  on  the  other 
side  of  E,  and  the  angle  C  A  E'  =  $  A  —  %  E'F'K.  It  is  obvious 
that,  in  both  cases,  it  E  G  is  exactly  equal  to  F  H,  the  angle 
E'F'K  vanishes,  and  C  A  E'  =  \  A. 


PARABOLIC    CURVES.  89 

CHAPTER  II. 

PARABOLIC   CURVES. 
ARTICLE  I. — LOCATING  PARABOLIC  CURVES. 

100.  LET  A  EB  (fig.  47)  be  a  parabola,  A  C  and  B  C  its  tan- 
gents, and  A  B  the  chord  uniting  the  tangent  points.  Bisect  A  B 
in  D,  and  join  C  D.  Then,  according  to  Analytical  Geometry, — 


Fig.  47. 


B 


I.  C  D  is  a  diameter  of  the  parabola,  and  the  curve  bisects  C  D 
in  E. 

II.  If  from  any  points  T,  T',  T",  &c.,  on  a  tangent  AF,  lines 
be  drawn  to  the  curve  parallel  to  tlie  diameter,  these  lines  T  M, 
T  M',  T"  M",  &c.,  called  tangent  deflections,  will  be  to  each 
other  as  the  squares  of  the  distances  A  T,  A  T',  A  T",  &c.,  from 
the  tangent  point  A. 

III.  A  line  E  D  (fig.  48),  drawn  from  the  middle  of  a  chord 
A  B  to  the  curve,  and  parallel  to  the  diameter,  may  be  called  the 
middle  ordinate  of  that  chord  ;  and  if  the  secondary  chords  A  E 
and  B  E  be  drawn,  the  middle  ordinates  of  these  chords,  KG  and 
L  H,  are  each  equal  to  %E  D.    In  like  manner,  if  the  chords  A  K, 
KE,  EL,  and  L  B  be  drawn,  their  middle  ordinates  will  be  equal 


IV.  A  tangent  to  the  curve  at  the  extremity  of  a  middle  ordi- 


90  PARABOLIC   CURVES. 

nate  is  parallel  to  the  chord  of  that  ordinate.    Thus  M F  (fig.  48), 
tangent  to  the  curve  at  E,  is  parallel  to  A  B. 

V.  If  any  two  tangents,  as  A  C  and  B  C  (fig.  48),  be  bisected  in 
M  and  F,  the  line  M  F,  joining  the  points  of  bisection,  will  be  a 
new  tangent,  its  middle  point  E  being  the  point  of  tangency. 

101.  Problem.  Given  the  tangents  A  C  and  B  (7,  equal  or 
unequal  (fig.  47).  and  the  chord  A  B,  to  lay  out  a  parabola  by 
tangent  deflections. 


Fig.  47. 


A  D  B 


Solution.  Bisect  A  B  in  D,  and  measure  C  D  and  the  angle 
A  CD;  or  calculate  CD*  and  ACD  from  the  original  data. 
Divide  the  tangent  A  C  into  any  number  n  of  equal  parts,  and 
call  the  deflection  T M  for  the  first  point  a.  Then  (§  100,  II.)  the 
deflection  for  the  second  point  will  be  T'  M'  =  4  &,  for  the  third 
point  T"  M "  =  9  a,  and  so  on  to  the  nth  point  or  (7,  where  it  will 
be  n2  a.  But  the  deflection  at  this  last  point  is  CE  =  $  CD  (§  100, 
I.).  Therefore,  n*a=CE,  and 

CE 

a  =  — r- 
n2 

Having  thus  found  a,  we  have  also  the  succeeding  deflections  4  a, 
9  a,  16  a,  &c.  Then  laying  oif  at  T7,  T7',  &c.,  the  angles  A  T  M, 
A  T'  M',  &c.,  each  equal  to  ACD,  and  measuring  down  the 
proper  deflections,  just  found,  the  points  M,  M',  &c.,  of  the  curve 
will  be  determined. 

*  Since  C  D  is  drawn  to  the  middle  of  the  base  of  the  triangle  A  B  (7,  we 
have,  by  Geometry,  CD*  =  ±(AC*  +  B  C*)  -  AD*. 


LOCATING   PARABOLIC    CURVES. 


91, 


The  direction  in  which  to  measure  the  deflections  may  be  ob- 
tained by  dividing  A  D  into  the  same  number  of  equal  parts  as 
A  C  and  joining  corresponding  points.  If  more  convenient  the 
chord  A  E  may  be  drawn,  and,  being  similarly  divided,  may  take 
the  place  of  A  D. 

The  curve  may  be  finished  by  laying  off  on  A  C  produced  n 
parts  equal  to  those  on  A  (7,  and  the  proper  deflections  will  be,  as 
before,  a  multiplied  by  the  square  of  the  number  of  parts  from  A. 
But  an  easier  way  generally  of  finding  points  beyond  E  is  to 
divide  the  second  tangent  B  C  into  equal  parts,  and  proceed  as  in 
the  case  of  A  C.  If  the  number  of  parts  on  B  C  be  made  the 
same  as  on  A  C,  it  is  obvious  that  the  deflections  from  both  tan- 
gents will  be  of  the  same  length  for  corresponding  points.  The 
angles  to  be  laid  off  from  B  C  must,  of  course,  be  equal  to  B  CD. 

The  points  or  stations  thus  found,  though  corresponding  to 
equal  distances  on  the  tangents,  are  not  themselves  equidistant. 
The  length  of  the  curve  is  obtained  by  actual  measurement  around 
the  stakes.  See  also  §  112. 

102.  Problem.  Given  the  tangents  A  C  and  B  C,  equal  or 
unequal  (fig.  4$},  and  the  chord  A  B,  to  lay  out  a  parabola  by 
middle  ordinates. 


Fig.  48. 


Solution.  Bisect  A  B  in  D,  draw  C  D,  and  its  middle  point  E 
will  be  a  point  on  the  curve  (§  100, 1.).  D  E  is  the  first  middle 
ordinate,  and  its  length  may  be  measured  or  calculated.  To  the 
point  E  draw  the  chords  A  E  and  B  E,  lay  off  the  second  middle 
ordinates  G  K  and  HL,  each  equal  to  ±  D  E  (§  100,  III.),  and  K 
and  L  are  points  on  the  curve.  Draw  the  chords  A  K,  KE,  EL, 
and  L  B,  and  lay  off  third  middle  ordinates,  each  equal  to  one 
fourth  the  second  middle  ordinates,  and  four  additional  points  on 


92  PARABOLIC    CURVES. 

the  curve  will  be  determined.     Continue  this  process,  until  a  suf- 
ficient number  of  points  is  obtained. 

103.  Problem.     To  draw  a  tangent  to  a  parabola  at  any 
station. 

Solution.  I.  If  the  curve  has  been  laid  out  by  tangent  deflec- 
tions (§  101),  let  M'"  (fig.  47)  be  the  station,  at  which  the  tangent 
is  to  be  drawn.  From  the  preceding  or  succeeding  station,  la\ 
off,  parallel  to  CD,  a  distance  M"  N  or  EL  equal  to  a,  the  first 
tangent  deflection  (§  101),  and  M'"  N  or  M' "  L  will  be  the  re- 
quired tangent.  The  same  thing  may  be  done  by  laying  off  from 
the  second  station  a  distance  M'  T'  =  4  a,  or  at  the  third  station 
a  distance  &  P  =  9  a ;  for  the  required  tangent  will  then  pass 
through  T'  or  6r.  It  will  be  seen,  also,  that  the  tangent  at  M'" 
passes  through  a  point  on  the  tangent  at  A  corresponding  to  half 
the  number  of  stations  from  A  to  M '"  ;  that  is,  M'"  is  four  sta- 
tions from  A,  and  the  tangent  passes  through  T',  the  second  point 
on  the  tangent  A  C.  In  like  manner,  M'"  is  six  stations  from  B, 
and  the  tangent  passes  through  6r,  the  third  point  on  the  tangent 
BC. 

II.  If  the  curve  has  been  laid  out  by  middle  ordinates  (§  102), 
the  tangent  deflection  for  one  station  is  equal  to  the  last  middle 
ordinate  made  use  of  in  laying  out  the  curve.  For  if  the  tangent 
A  C  (fig.  48)  were  divided  into  four  equal  parts  corresponding  to 
the  number  of  stations  from  A  to  E,  the  method  of  tangent  de- 
flections would  give  the  same  points  on  the  curve,  as  were  ob- 
tained by  the  method  of  §  102.  In  this  case  the  tangent  deflec- 
tion for  one  station  would  be  a  =  -^  C  E  —  ^6D  E\  but  the  last 
middle  ordinate  was  made  equal  to  £  Gr  K  or  ^  D  E.  Therefore, 
a  is  equal  to  the  last  middle  ordinate,  and  a  tangent  may  be 
drawn  at  any  station  by  the  first  method  of  this  section. 

A  tangent  may  also  be  drawn  at  the  extremity  of  any  middle 
ordinate,  by  drawing  a  line  through  this  extremity,  parallel  to  the 
chord  of  that  ordinate  (§  100,  IV.). 

104.  In  laying  out  a  parabola  by  the  method  in  §  101,  it  may 
sometimes  be  impossible  or  inconvenient  to  lay  off  all  the  points 
from  the  original  tangents.     A  new  tangent  may  then  be  drawn 
by  §  103  to  any  station  already  found,  as  at  M'"  (fig.  47),  and  the 
tangent  deflections  a,  4  a,  9  a,  &c.,  may  be  laid  off  from  this  tan- 
gent, precisely  as  from  the  first  tangent.     These  deflections  must 


LOCATING   PARABOLIC    CURVES.  93 

be  parallel  to  C  D,  and  the  distances  on  the  new  tangent  must  be 
equal  to  T'N  or  N M' ",  which  may  be  measured. 

105.  Problem.  Given  the  tangents  A  C  and  B  C,  equal  or 
unequal  (fig.  49),  to  lay  out  a  parabola  by  bisecting  tangents. 

Solution.  Bisect  A  C  and  B  C  in  D  and  F,  join  D  F,  and  find 
E,  the  middle  point  of  D  F.  E  will  be  a  point  on  the  curve 
(J5  100,  V.).  We  have  now  two  pairs  of  what  may  be  called  second 
tangents,  A  D  and  D  E,  and  EF  and  FB.  Bisect  A  D  in  G  and 
7)  E  in  H,  join  G  H,  and  its  middle  point  M  will  be  a  point  on 


Fig.  49. 


the  curve.  Bisect  E  F  and  F  B  in  K  and  L,  join  KL,  and  its 
middle  point  N  will  be  a  point  on  the  curve.  We  have  now  four 
pairs  of  third  tangents,  A  O  and  G  M,  ME  and  HE,  E TTand 
K  N,  and  N  L  and  L  B.  Bisect  each  pair  in  turn,  join  the  points 
of  bisection,  and  the  middle  points  of  the  joining  lines  will  be  four 
new  points,  M',  M",  N".  and  N'.  The  same  method  may  be  con- 
tinued, until  a  sufficient  number  of  points  is  obtained. 

106.  Problem.  Given  the  tangents  A  C  and  B  C,  equal  or 
unequal  (fig.  50),  and  the  chord  A  B,  to  lay  out  a  parabola  by 
intersections. 

Solution.  Bisect  A  B  in  Z>,  draw  CD,  and  bisect  it  in  E. 
Divide  the  tangents  A  C  and  B  C,  the  half-chords  A  D  and  D  B, 
and  the  line  C  E,  into  the  same  number  of  equal  parts;  five,  for 
example.  Then  the  intersection  M  of  A  a  and  F  Gr  will  be  a  point 
on  the  curve.  For  FM—^Ca,  and  Ca  —  ^CE.  Therefore, 
F  M  =  -£%  C  E,  which  is  the  proper  deflection  from  the  tangent  at 
F  to  the  curve  (§  101).  In  like  manner,  the  intersection  N  of  A  b 
and  HKwsij  be  shown  to  be  a  point  on  the  curve,  and  the  same 
is  true  of  all  the  similar  intersections  indicated  in  the  figure. 


94  PARABOLIC    CURVES. 

If  the  line  D  E  were  also  divided  into  five  equal  parts,  the  line 
A  a  would  be  intersected  in  M  on  the  curve  by  a  line  drawn  from 
B  through  a',  the  line  A  b  would  be  intersected  in  N  on  the  curve 


Fig.  50. 


by  a  line  drawn  from  B  through  b',  and  in  general  any  two  lines, 
drawn  from  A  and  B  through  two  points  on  C  D  equally  distant 
from  the  extremities  C  and  Z>,  will  intersect  on  the  curve.  To 
show  this  for  any  point,  as  Jl,  it  is  sufficient  to  show,  that  B  a' 
produced  cuts  F  Gr  on  the  curve ;  for  it  has  already  been  proved, 
that  A  a  cuts  F  G  on  the  curve.  Now  Da1 :  MG  =  B  D  :  B  Gr  = 
5  :  9,  or  M  G  =  f  D  a'.  But  D  a'  =  J-  C  E.  Therefore,  M  G  = 
G:CD  =  Aa:AD  =  l:5.  Therefore,  F  Q  = 
We  have  then  FM=FG  — MG  =  $CE  — 
•£F  C  E  =  -fa  C  E.  As  this  is  the  proper  deflection  from  the  tan- 
gent at  F  to  the  curve  (§  101),  the  intersection  of  B  a'  with  F  G 
is  on  the  curve.  This  furnishes  another  method  of  laying  out  a 
parabola  by  intersections. 

107.  The  following  example  is  given  in  illustration  of  several 
of  the  preceding  methods. 

Example.  Given  A  C  =  B  C  -  832  (fig.  51),  and  AB  =  1536, 
to  lay  out  a  parabola  A  E  B.  We  here  find  CD  —  320.  To  be- 
gin with  the  method  by  tangent  deflections  (g  101),  divide  the 

C1  W       1  fif) 
tangent  A  C  into  eight  equal  parts.     Then  a  —  — —  —  -^r  =  2.5. 

Lay  off  from  the  divisions  on  the  tangent  F 1  =  2.5,  G  2  =  4  x 
2.5  =  10,  HZ  =  9  x  2.5  =  22.5,  and  K±  =  16  x  2.5  =  40.     Sup- 


RADIUS   OF   CURVATURE. 


95 


pose  now  that  it  is  inconvenient  to  continue  this  method  beyond 
K.  In  this  case  we  may  find  a  new  tangent  at  JE,  by  bisecting 
A  C  and  B  C  (§  105),  and  drawing  KL  through  the  points  of  bi- 
section. Divide  the  new  tangent  K E  =  -J-  A  D  =  384  into  four 
equal  parts,  and  lay  off  from  K  E  the  same  tangent  deflections  as 
were  laid  off  from  A  K,  namely,  M  5  =  22.5,  N6  =  10,  and  01  — 


Fig.  51. 


2.5.  To  lay  off  the  second  half  of  the  curve  by  middle  ordinates 
(§  102),  measure  EB  =  784.49.  Bisect  E  B  in  P,  and  lay  off  the 
middle  ordinate  PR  =  £  D  E  =  40.  Measure  E  R  =  386.08,  and 
B  Rs=  402.31,  and  lay  off  the  middle  ordinates  S  T  and  V  W,  each 
equal  to  %PR  =  10.  By  measuring  the  chords  E  T,  T  R,  RW, 
and  W B,  and  laying  off  an  ordinate  from  each,  equal  to  2.5,  four 
additional  points  might  be  found. 


ARTICLE  II.— RADIUS  OF  CURVATURE. 

108.  THE  curvature  of  circular  arcs  is  always  the  same  for  the 
same  arc,  and  in  different  arcs  varies  inversely  as  the  radii  of  the 
arcs.  Thus,  the  curvature  of  an  arc  of  1,000  feet  radius  is  double 
that  of  an  arc  of  2,000  feet  radius.  The  curvature  of  a  parabola 
is  continually  changing.  In  fig.  50,  for  example,  it  is  least  at  the 
tangent  point  A,  the  extremity  of  the  longest  tangent,  and  in- 
creases by  a  fixed  law,  until  it  becomes  greatest  at  a  point,  called 
the  vertex,  where  a  tangent  to  the  curve  would  be  perpendicular 
to  the  diameter.  From  this  point  to  B  it  decreases  again  by  the 


96  PARABOLIC    CURVES. 

same  law.  We  may,  therefore,  consider  a  parabola  to  be  made  up 
of  a  succession  of  infinitely  small  circular  arcs,  the  radii  of  which 
continually  increase  in  going  from  the  vertex  to  the  extremities. 
The  radius  of  the  circular  arc,  corresponding  to  any  part  of  a 
parabola,  is  called  the  radius  of  curvature  at  that  point. 

If  a  parabola  forms  part  of  the  line  of  a  railroad,  it  will  be  ne- 
cessary, in  order  that  the  rails  may  be  properly  curved  (§  28),  to 
know  how  the  radius  of  curvature  may  be  found.  It  will,  in  gen- 
eral, be  necessary  to  find  the  radius  of  curvature  at  a  few  points 
only.  In  short  curves  it  may  be  found  at  the  two  tangent  points 
and  at  the  middle  station,  and  in  longer  curves  at  two  or  more 
intermediate  points  besides.  The  rails  curved  according  to  the 
radius  at  any  point  should  be  sufficient  in  number  to  reach,  on 
each  side  of  that  point,  half-way  to  the  next  point. 

109.  Problem.  To  find  the  radius  of  curvature  at  certain 
stations  on  a  parabola. 

Solution.  Let  A  E  B  (fig.  52)  be  any  parabola,  and  let  it  be  re- 
quired to  find  the  radii  of  curvature  at  a  certain  number  of  sta- 
tions from  A  to  E.  These  stations  must  be  selected  at  regular 


Fig.  52. 


intervals  from  those  determined  by  any  of  the  preceding  methods. 
Let  n  denote  the  number  of  parts  into  which  A  E  is  divided,  and 
divide  CD  into  the  same  number  of  equal  parts.  Draw  lines 
from  A  to  the  points  of  division.  Thus,  if  n  =  4,  as  in  the  figure, 
divide  C  D  into  four  equal  parts,  and  draw  A  F,  A  E,  and  A  G. 
LetAD  =  c,  AF  =  c1,AE=c*,A  G  =  c3,&ndAC  =  T.  De- 
note, moreover,  CD  by  d,  and  the  area  of  the  triangle  A  C  B  by 


RADIUS   OF   CURVATURE.  9? 

A.  Then  the  respective  radii  for  the  points  E,  1,  2,  3,  and  A 
will  be 

7?        °3         7?        ClB         7?        C'3         7?   -  Ca*         7?         ^3 

^  =  z>   ^Z'    •*   4'   ^'-Z'   ^4-T- 

The  area  J.  may  be  found  by  form.  18,  Tab.  X.  ;  c  and  T  are 
known  ;  and  Ci,  ca,  c3  may  be  found  approximately  by  measure- 
ment on  a  figure  carefully  constructed,  or  exactly  by  these  gen- 
eral formulae  :  — 


2  =  c2   , 


c  t        s  +  T*-c*  _  (n  -  3)  d« 
n  n* 

c  2  _  c  2  +  r*  ~  c2  _  (n  ~  5)  ^2 

71  7l2  ' 

2  -     2      T*  -c*  _  (n-T)d* 
C*  ~~  °*  n  7i2       ' 

&c.,  &c. 

It  will  be  seen,  that  each  of  these  values  is  formed  from  the  pre- 

^2 C2  ^2 

ceding,  by  adding  the  same  quantity  —       —  ,  and  subtracting  -^ 

multiplied  in  succession  by  n  —  1,  n  —  3,  n  —  5,  &c.    Making  n  = 
4,  we  have 


All  the  quantities,  which  enter  into  the  expressions  for  the  radii, 
are  now  known,  and  the  radii  may,  therefore,  be  determined.  The 
same  method  will  apply  to  the  other  half  of  the  parabola. 

The  manner  of  obtaining  the  preceding  formulae  is  as  follows : 
The  radius  of  curvature  at  any  given  point  on  a  parabola  is,  by 

the  Differential  Calculus,  R  =      .    8     ,  in  which  p  represents  the 

parameter  of  the  parabola  for  rectangular  coordinates,  and  E  the 
angle  made  with  a  diameter  by  a  tangent  to  the  curve  at  the  given 
point.  First,  let  the  middle  station  E  (fig.  53)  be  the  given  point. 
Then  the  angle  E  is  the  angle  made  with  E  D  by  a  tangent  at  E, 
or  since  A  B  is  parallel  to  the  tangent  at  E  (§  100,  IV.),  sin.  E  = 
sin.  A  D  E  =  sin.  B  D  E.  Let  p'  be  the  parameter  for  the  diam- 


98 


PARABOLIC    CURVES. 


eter  ED.    Then,  by  Analytical  Geometry,  p  =p'  sin.2^.     There- 


fore, at  this  point  R  — 


=  •         '        =  —  ~ 


A    7)2  r<i 

•"-  •*-*'  &  mi  £  T~> 

^  _.  =  —3  .    Therefore,  R  — 
ED       id 


—  ~  —  =,  .    But  p'  = 
2  sin.  E 

/.2  ,,%  „% 

C/  . 

—  —  -  —  —=—.  —  —  =  —  ;  since  A  — 
dsin.fi      cdsm.fi      A 


c  dsin.fi  (Tab.  X.  17). 

Next,  to  find  Ri  ,  or  the  radius  of  curvature  at  H,  the  first  sta- 
tion from  E.     Through  IT  draw  F  G  parallel  to  CD,  and  from  F 


Fig.  53. 


draw  the  tangent  F  K.    Join  A  K,  cutting  CD  in  L.    Then  from 
what  has  just  been  proved  for  the  radius  of  curvature  at  E,  we 

A     z 
have  for  the  radius  of  curvature  at  H,  RI  — 


.  77  „. 

A.  JJ  J\. 


Now  A  O  : 


For,  since  AF  — 


=  n-l:n,  ov 

— 


—  1 


x  AL. 


x  AC,  the  tangent  deflection  FH= 


•      (§  100,  II.),  and  F  G  =  2  FH  = 


Then. 


-l,  C  L  = 


d.    Hence  L  D  =  d- 


_  1  1  n  —  I 

—  d=-  d,  that  is,  ^  ^  ~  / 
^  n  n 

Substituting  this  value  in  the  expression  for  A  Gr  above,  we  I  . 


A  Gr  — 


Moreover,  since  AF=  -  x  A  C,  and 


n  n 

cause  similar  triangles  are  to  each  other  as  the  squares  of  their 


homologous  sides,  we  have  the  triangle  A  F  G  = 


x  ACL. 


But  ACL:  A  C  D=  C  L:  CD  =  n-  1  :  n,  or  A  CL  = 


RADIUS    OF   CURVATURE.  99 


x  A  CD.    Therefore,  A  F  G  =      ~       x  A  CD,  and  A  FK  = 


2AFG  =  (n  ~  1)S  x  A  C  B  =  ^  ~  1)3  A.    Substituting  these 

if  1  6  /no 

A  G* 

values  of  A  G  and  A  F  K  in  the  equation  Rl  =  AFK'  an(*  re" 

ducing,  we  find  R\  =  -j~  .     By  similar  reasoning  we  should  find 


It  remains  to  find  the  values  of  e,,  ca,  &c.  Through  A  draw 
A  M  perpendicular  to  CD,  produced  if  necessary.  Then,  by  Ge- 
ometry, we  have  AD*  =  AL*  +  LD*-2LD  x  LM,  and  AC9 
—  A  L*  +  C  L*  +  2  C  L  x  L  M.  Finding  from  each  of  these 
equations  the  value  of  2LM,  and  putting  these  values  equal  to 

AL*  +  LD*-AD*      AW-AL*-  C  L* 
each  other,  we  have  -  ^r—  -  —  —  -^-=.  —        -  . 

But  AL  =  d,  LD=-d,AD  =  c,  AC  =  T,  and  CL  =  ^^  d. 
n  n 

Substituting  these  values  in  the  last  equation,  and  reducing,  we 
find 

f  =  T^      (n-l)c*  _  (n-l)d^ 

n  n  n? 

By  similar  reasoning  we  should  find 

_2T*      (n-2)c*      2(n-2)d* 

C%      -    -      +     -      --  a  -    5 

n  n  n1 


a  , 

n  n* 

&c.,  &c. 

From  these  equations  the  values  of  Ci2,  c22,  c32,  &c.,  given  above, 
are  readily  obtained.  That  given  for  Ci2  is  obtained  from  the  first 
of  these  equations  by  a  simple  reduction  ;  that  given  for  ca2  is  ob- 
tained by  subtracting  the  first  of  these  equations  from  the  second, 
and  reducing  ;  that  given  for  cs2  is  obtained  by  subtracting  the 
second  equation  from  the  third,  and  reducing  ;  and  so  on. 

110.  Example.  Given  (fig.  52)  A  C  =  T  =  600,  B  C  =  T'  = 
520,  and  A  D  =  c  —  550,  to  find  R,  R^  ,  J?2  ,  7?3  ,  and  jK4  .  the  radii 
of  curvature  at  E,  1,  2,  3,  and  A. 

To  find  CD  =  d,  we  have,  by  Geometry,  d9  =  i(T*  +  T'2)  - 
c2  which  gives  d*  =  12700. 


100  PARABOLIC    CURVES. 

To  find  the  area  of  A  C  B  =  A,  we  have  (Tab.  X.  18)  A  = 
—  a)(s  —  b)  (s  —  c). 


8=  1110 

3.045323 

s  — 

a  =  590 

2.770852 

s  — 

6  =  510 

2.707570 

8  — 

c  =  10 

1.000000 

2)9.523745 

log.  A  4.761872 

Next  1  (T2  -  c»)  =  i  (T  +  c)  (T-c)  =  115°4X  5°  =  14375,  and 
Then 


n*    lo 

c2  =  5502  =  302500 

ci»  =  302500  +  14375  -  3  x  793.75  =  314493.75 
ca2  =  314493.75  +  14375  -  793.75  =  328075 
ca2  =  328075  +  14375  +  793.75  =  343243.75. 

c3 

To  find  R,  we  have  R  =  -?  ,  or  log.  R  =  3  log.  c  —  log.  A, 
A 

c  =  550  2.740363 


c3  8.221089 

A  4.761872 


R  =  2878.8  3.459217 


To  find  Ri  ,  we  have  J?i  =  -j-  ,  or  log.  J£i  =  -  log.  Ci2  —  log.  A, 

Cl2  =  314493.75  5.497612 


d8  8.246418 

A  4.761872 


J2i  =  3051.7  3.484546 

In  the  same  way  we  should  find  R*  =  3251.5,  R3  =  3479.6,  R*  - 
3737.5. 

To  find  the  radii  for  the  second  part  E  B  of  the  parabola,  the 
same  formula?  apply,  except  that  T'  takes  the  place  of  T.  We 


RADIUS   OF   CURVATURE.  101 

have  then  -  (T'2  -  c2)  =  ±  (T1  +  e)  (T1  -  c)  =  107°  *  ~  3Q  = 
n  4 

-  8025.     Hence 

c12  =  302500  -  8025  -  2381.25  =  292093.75. 
c22  =  292093.75  -  8025  -  793.75  =  283275. 
c32  =  283275  -  8025  +  793.75  =  276043.75. 

c  3  S 

To  find  ^j,  we  have  Rl  =  -|- ,  or  log.  ^1=7;  log.  d2  —  log.  4, 
-A  tit 

d2  =  292093.75  5.465523 

d3  8.198284 

A  4.761872 


7?!  =  2731.6  3.436412 

In  the  same  way  we  should  find  R*  =  2608.8,  R3  =  2509.5,  R4  = 
2433. 

It  will  be  seen  that  the  radii  in  this  example  decrease  from  one 
tangent  point  to  the  other,  which  shows  that  both  tangent  points 
lie  on  the  same  side  of  the  vertex  of  the  parabola  (§  108).  This 
will  be  the  case,  whenever  the  angle  BCD,  adjacent  to  the  shorter 
tangent,  exceeds  90°,  that  is,  whenever  c2  exceeds  T'2  +  d?.  If 
B  CD  =  90°,  the  tangent  point  B  falls  on  the  vertex.  It  BCD 
is  less  than  90°,  one  tangent  point  falls  on  each  side  of  the  ver- 
tex, and  the  curvature  will,  therefore,  decrease  towards  both  ex- 
tremities. 

111.  If  the  tangents  Tand  T'  are  equal,  the  equations  for  d2, 
c22,  &c.,  will  be  more  simple ;  for  in  this  case  d  is  perpendicular  to 
c.  and  T2  —  c2  =  c?2.  Substituting  this  value,  we  get 


=  d2  +  - 


&c.,        &c. 

Example.     Given,  as  in  §  107,  T—  T'  =  832,  c  =  768,  and  d  = 
320,  to  find  the  radii  7?,  R^ ,  and  R*  at  the  points  E,  4,  and  A  (fig. 


102 


PARABOLIC     CURVES. 


51).    Here  A  =  c  d  =  245760,  n  =  2,  and  cj  =  c*  + 

Then  R  =  ^  =  ^=  ^  =  1843.2,  ^  =  —, ,  and  . 
cd       d       320  cd 

d2  =  615424 


c  d  =  245760 
fit  =  1964.5 


cd  =  245760 
^2  =  2343.5 


*  =  615424. 


5.789174 

8.683761 
5.390511 

3.293250 
2.920123 

8.760369 
5.390511 


3.369858 

Ri  is  the  radius  at  the  point  R  also,  and  J?a  the  radius  at  the 
point  B. 
112.  Length  of  parabolic  arcs. 


B 


The  length  s  of  the  parabolic  arc  A  B  (fig.  54)  from  the  vertex 
A  to  a  point  B  whose  rectangular  coordinates  are  x  and  y  is,  by 
the  Calculus, 


or,  introducing  the  angle  i  which  the  tangent  at  B  makes  with 
the  axis  of  x, 

x2 
s  =  -£-  [tan.  i  sec.  i  +  hyp.  log.  (tan.  i  +  sec.  *)]  ; 

or,  by  series, 


LENGTH  OF  PARABOLIC  ARCS. 


103 


When  y  is  small  relatively  to  x,  two  terms  of  this  series  are  often 

sufficient.    Whence 

2y* 

s  =  x  -H  Q  —  nearly, 
o  x 

The  length  s  of  the  parabolic  arc  A  B  (fig.  55)  from  the  origin 
of  oblique  coordinates  A  to  a  point  B  whose  oblique  coordinates 
are  x  and  y,  is  given  by  the  following  formula,  in  which  i  is  the 


Fig.  55. 


angle  made  by  the  tangent  at  B  with  a  line  perpendicular  to  the 
axis  of  the  parabola,  and  j  is  the  angle  made  by  y  with  a  perpen- 
dicular to  the  axis  A  X. 

x*  cos.2//,        .         .  •     i        i       tan.  i  +  sec.  i\ 

s  =  — - — =  ( tan.  i  sec.  i  —  tan.  j  sec.  j  +  hyp.  log. : —     — . ) 

4  y     \  to  tan.^  +  sec.j/' 

In  many  cases  a  near  approximation  is 

2      y2  cos.2/ 


s  =  x  +  y  sm.j  +  ^  • 


3    x  +  ysiu.j" 


104  TRANSITION    CURVES. 

CHAPTER  III. 

TRANSITION  CURVES. 

113.  THE  object  of  a  transition  curve  is  to  make  the  change 
easy  from  a  straight  line  to  a  circular  curve.    The  proper  super- 
elevation of  the  outer  rail  of  the  circular  curve  is  also  arrived  at 
by  a  gradual  rise  from  the  straight  line.     To  make  this  rise  uni- 
form, the  radius  of  curvature  of  the  transition  curve  must  be  in- 
finite at  its  beginning  on  the  straight  line,  must  decrease  in  such 
a  way  that,  at  any  point  of  the  curve,  it  shall  be  inversely  as  the 
distance  of  that  point  from  the  beginning,  and,  finally,  become 
equal  to  the  radius  of  the  circular  curve,  where  it  joins  that  curve 
tangentially.    The  cubic  parabola  fulfils  all  the  essential  requi- 
sites of  such  a  transition  curve.    The  compound  circular  curve 
(§  132)  forms  another  method  of  easing  the  change  from  a  straight 
line  to  a  circular  curve. 

ARTICLE  I.  —  THE  CUBIC  PARABOLA. 

114.  Let  GDC'  (fig.  56)  be  the  central  circular  curve  of  radius 
0  C  =  R.    Let  ABC  and  A'  B'  C'  be  the  transition  curves,  con- 
necting the  circular  curve  with  the  tangents  at  A  and  A  '.    Let  x 
and  y  be  the  rectangular  coordinates  of  A  B  C,  with  origin  at  A, 
and  let  Xi  and  yi  denote  the  coordinates  of  the  point  C.    Let  the 
rise  of  the  outer  rail  be  taken  as  uniform  for  distances  from  A 
along  the  axis  of  x,  instead  of  along  the  curve,  an  immaterial 

change,  and  let  -.  denote  the  rate  of  rise.     Then  the  rise  at  any 
i/ 

distance  x  from  A  will  be  -  .    This  rise  may  be  expressed  in  an- 

other way.     For  let  p  denote  the  radius  of  curvature  of  the  curve 
at  the  point  whose  abscissa  is  x,  and  we  have  the  rise  e  by  the  for- 

mula of  §  152,  e  —    9V    .    Equating  the  two  values, 
o2.53  p 


q  v*  i 

'=&!*• 


THE    CUBIC    PARABOLA. 


105 


When  the  velocity  v  has  been  fixed,  and  also  the  rate  of  rise  -7  , 
the  quantity  ~7  becomes  a  constant.    At  (7,  the  radius  of  curva- 


ture  p  becomes  R,  and  x  becomes  Xi  ,  so  that  equation  (1)  becomes 


and  we  have  7^7  =  R  %i  .    By  substitution  (1)  becomes 


Another  expression  for  p  is,  by  the  Differential  Calculus, 


106  TRANSITION   CURVES. 

where  d  s  is  the  differential  of  the  length  of  the  curve.  In  the 
present  case,  the  differential  d  x  of  the  abscissa  is  so  nearly  equal 
to  d  s,  that  we  may  put 

dx*    ^dtf 
P~  dxd*y~~  d*y' 

Equating  the  two  values  of  p,  and  inverting,  we  have 
d*y  _    x 
dz?~  Rx^ 
Integrating  once,  we  have 

f*y 
dx 

and,  integrating  again, 


115.  This  is  the  equation  of  a  cubic  parabola  —  that  is,  of  a  curve 
in  which  the  ordinates  are  proportional  to  the  cubes  of  the  ab- 
scissas.    The  curves  ABC  and  A'  B  '  C'  are,  therefore,  to  be 
treated  as  cubic  parabolas.    Before  doing  this,  however,  two  prob- 
lems require  consideration.     For  in  order  to  connect  two  straight 
lines  or  tangents,  as  A  1  and  A'  I,  by  a  central  circular  curve,  with 
a  transition  curve  at  each  end,  we  have  either  to  find  A  I  =  Ty 
when  the  radius  0  C  =  R  of  the  circular  curve  is  given,  or  to  find 
R,  when  T  is  given.    In  both  cases  the  intersection  angle  I  is 
supposed  to  be  known,  and  the  value  of  Xi  =  A  E  to  be  assumed. 

116.  Problem.      Given   the  intersection  angle  1=2  G  01 
(fig.  56\  the  abscissa  xlt  and  the  radius  0  C  =  R  of  the  central 
curve,  to  find  the  tangent  AI=T. 

Solution.  In  the  figure  the  circular  curve  is  produced  to  <?, 
where  its  tangent  becomes  parallel  to  A  I.  Draw  0  G  and  pro- 
duce it  to  H.  Draw  also  C  F,  the  common  tangent  at  (7,  and 
C  K  parallel  to  A  I.  Denote  the  angle  C  0  G  -  C  FE  by  A.  To 
find  T  we  have 

T=AH+  HI. 

Now  AH-A  E-  H  E  =  xl-H  E  =  xl-C  K  =  xl-R 
sin.  A. 

But,  since  the  angle  A  is  generally  small,  we  may  put  sin.  A  = 
tan.  A,  and  we  have 

AH=Xi  —  R  tan.  A. 


THE   CUBIC   PARABOLA.  107 

Now  ft  tan.  A  =  £  x^  .    For  by  the  Differential  Calculus  we  know 

that  -~  in  equation  (2)  denotes  the  tangent  of  the  angle  made 
a  x 

with  the  axis  of  #  by  a  tangent  to  the  curve  at  a  point  whose  ab- 
scissa is  x.  Now  when  the  abscissa  becomes  Xi  at  the  point  C, 
this  angle  becomes  C  F  E  —  A,  and  we  have 

tan-  A  =  *  =      =    >  and  R  tan-  A  =  ia;-: 


.  *  .  A  H  =  Xi  —  -J-  Xi  =  £  #1  .* 
Next  to  find  ZT/,  we  have 

#/=:  OHtsiu.iI=(R  +  0IT)  tan.i/. 

Gr  His  the  perpendicular  distance  between  the  tangent  A  E  and  a 
tangent  to  the  circular  curve  at  G.  This  is  usually  called  the 
shift,  and  may  be  denoted  by  s.  To  find  Cr  H  —  s  we  have  5  = 
CE-GK=yl-GK.  By  equation  (3) 

fti3     _x^_ 
yi  '    6  R  x,      6  72  ' 
and  G  Kis  the  middle  ordinate  of  the  circular  curve  for  a  chord 

=  xl.    Therefore,  (§  26),  G  K  =  ^4r  5  so  that 

O  -ft 


Substituting  this  value  ofs=GH  in  the  equation  for  H  J,  we 
have 

JJ  /=  CK  +  i  yi)  tan.  i  7. 

Finally,  substituting  the  values  found  for  A  H  and  HI  in  the 
equation  for  T,  we  have 

T=%xl  +  (R  +  i 


117.  Problem.  ^iVew  the  intersection  angle  I  —  2  Gr  0 1 
(fig.  56\  the  abscissa  x^  and  the  tangent  AI=  T,  to  find  the  ra- 
dius 0  C  =  R  of  the  circular  curve. 


*  When  thought  necessary,  A  Hm&y  be  calculated  accurately  by  the  for- 
mula AH  —  z\  -  Rsin.  A. 

t  The  formula  GK=R(\  —  cos.  A)  gives  the  exact  value  of  G  K,  but  the 
difference  is  generally  unimportant. 


108  TRANSITION   CURVES. 

Solution.     From  the  preceding  section  we  have 


Compute  this  value  of  R  +  £3/1,  and  from  it  subtract  an  assumed 
probable  value  of  £  3/1  .     This  will  give  an  approximate  value  of 

x  2 
R,  and  with  this  compute  £3/1  by  the  formula  i^/i  —  5-™.     If 


the  value  so  found  agrees  nearly  enough  with  the  assumed  value 
of  £  2/1  ,  the  approximate  value  of  R  may  be  taken  as  the  true 
value.  Otherwise,  a  new  approximation  is  to  be  computed.  Gen- 
erally, however,  the  value  of  R  thus  found  would  be  used  only  to 
select  a  convenient  deflection  angle  for  the  central  curve.  The 
corresponding  value  of  R  may  then  be  used  to  find,  by  section  116, 
a  new  value  of  T.  A  change  in  the  value  of  T  would  of  course 
change  the  position  of  the  tangent  point,  but  seldom  materially. 
118.  Length  of  the  abscissa  Xi  .  Let  us  now  consider  the  value 

to  be  given  to  x\  .    The  rate  of  rise  of  the  outer  rail  being  -  ,  the 
total  rise  at  the  end  of  the  transition  curve  will  be  -^  .    This  total 

rise  is  also  expressed  by  e  =  Q(f  0  p  (§  152).    Equating  these  values, 
' 


we  have  —  =  e,  or  Xi  =  i  e.    The  length  of  Xi  is,  therefore,  depend- 
i 

ent  on  i  and  e.  The  value  of  i  may  be  taken  as  varying  from  300 
to  600,  corresponding  to  grades  of  17.6  feet  to  8.8  feet  per  mile. 
The  value  of  e  depends  upon  the  velocity  of  trains  and  the  radius 
of  the  curve.  For  high  speeds  e  may  vary  from  e  =  .3  to  e  —  .5. 
A  value  of  e  =  .5  allows  a  speed  of  67  miles  per  hour  on  a  2° 
curve,  of  30  miles  per  hour  on  a  10°  curve,  and  of  25  miles  per 
hour  on  a  14°  curve ;  so  that  this  value  of  e  would  rarely  be  ex- 
ceeded. With  i  =  300,  Xi  need  not  exceed  150  feet,  and  with  i  = 
600,  xl  need  not  exceed  300  feet.  These  lengths  might  of  course 
in  exceptional  cases  be  increased. 

119.  Let  the  length  of  Xi  be  expressed  in  rail  lengths  of  30  feet 
each,  and  let  n  denote  the  number  of  such  rail  lengths.  We  shall 
then  have 

xl  =  3Qn. 

x  3 

To  express  yi ,  we  have  from  equation  (3)  yi  =  n-^— 

o  a  x\ 


THE    CUBIC    PARABOLA. 


109 


?00n»  =  1BO«» .    substituting  for  R  its  value,  R  =  -^ ,  D  be- 
6  It  ±t  sin.  u 

ing  the  deflection  angle  of  the  circular  curve  for  chords  of  100 

150  n2  sin.  D 

feet,  we  have  y\  = r^-  — ,  or 

ou 

yl  =  3  n2  sin.  D. 


Fig.  56. 
To  fix  the  position  of  the  common  tangent  C  F,  we  require  the 


distance  F  E.    The  triangle  CF  E  gives  FE= 


i 
(§  116)  tan.  A  =  —  --  = 


30  n      30  7i  sin.  D 


' 


,  and  by 


— 

lUU 


0      .      ^     0  ,    ... 
=  .3  n  sm.  D.    Substitut- 


ing  this  value  and  that  of  7/1  ,  we  have 
3fi2sin.  D       . 


,.,  ^ 

FE  = 


^—.  —  =. 

.3  n  sm.  D 


110 


TRANSITION    CURVES. 


120.  Method  by  Offsets.  With  R  or  /),  T,  xl ,  and  y,  known, 
the  curves  can  now  be  laid  out.  J.,  the  point  of  beginning  or 
origin,  is  a  fixed  point,  from  which  xl  =  30  n  is  measured  to  fix 
the  point  E ;  yl=3n'2  sin.  D  fixes  the  point  C ;  and  F  E  =  $  Xi  = 
10  n  fixes  the  position  of  the  common  tangent  C  F.  Intermediate 
points  on  the  transition  curve  are  fixed  by  offsets  or  ordinates 
from  the  tangent  A  E,  thus :  divide  A  E  into  n  equal  parts  and 
denote  the  successive  offsets  at  the  points  of  division  by  di ,  d* ,  d* , 
•  •  •  •  dn .  Then  dn  =  yi ,  and,  as  the  ordinates  are  as  the  cubes  of 

7        Vi       3  n2  sin.  D      3  sin.  D       m, 

the  abscissas,  di  =  —a  = = .     The  successive 

-    ,  n9  nz  n 

offsets  are  then 


#1=^!,    da  =  8  d, ,    dj  =  27  di , 


•  dn  =  yi . 


The  circular  curve  CDC'  is  now  run  in  the  usual  way  from 
the  tangent  C  F  produced,  with  D  as  the  deflection  angle  for  100 
feet  chords.  The  central  angle  of  this  carve  is  COC'  =  I—2&. 
At  C',  E'C1  should  prove  equal  to  yit  The  distance  D  I  is  equal 
to  the  ordinary  external  D  L,  increased  by  L  I  =  (r  //sec.  i  I  — 
\yi  sec.  £/.  The  second  transition  curve  A'B'C'  is  the  same  as 
ABC  reversed,  and  is  laid  out  in  the  same  way. 

121.  The  annexed  table  gives  the  necessary  data  for  curves  from 
60  to  300  feet  in  length.  D  is  the  deflection  angle  of  the  central 
curve  for  100  feet  chords.  For  any  other  chord  c  it  is  only  neces- 

100 

sary  to  multiply  the  values  given  for  yi  and  di  by  —  .    Thus  if 

c 

D  were  the  deflection  angle  for  50  feet  chords,  we  should  have 
y^  —  6  ft2  sin.  D  and  di  =  —  — .  In  computing  y-i  and  di  use  nat- 
ural sines. 

TABLE  A. 


n 

„                  Ort   .. 

y^  =  3  na  sin.  D 

3  sin.  D 

dl           n 

2 

60 

12  sin.  D 

1.5  sin.  D 

3 

90 

27  sin.  D 

1.  sin.  D 

4 

120 

48  sin.  D 

.75  sin.  D 

5 

150 

75  sin.  D 

.6  sin.  D 

6 

180 

108  sin.  D 

.5  sin.  D 

7 

210 

147  sin.  D 

f  sin.  D 

8 

240 

192  sin.  D 

f  sin.  D 

9 

270 

243  sin.  D 

i  sin.  /) 

10 

300 

300  sin.  D 

.3  sin.  /> 

THE    CUBIC    PARABOLA.  Ill 

It  will  be  seen  that  this  method  applies  directly,  whether  the 
central  curve  is  of  an  even  degree  or  not,  since  sin.  D  may  be 
taken  from  the  table  for  any  value  of  Z>. 

122.  Example,  when  R  or  D  is  given.    Given  /  =  72°  40',  D  = 
3°  20',  and  n  =  8.     Here  x,  =  240,  yl  =  192  sin.  3°  20'  =  192  x 
.05814  =  11.16288.     From  Table  I.,  R  =  859.92,  and  ±yi  =  2.79. 

First  find  T. 

R  +  iy1==  862.71  2.935865 

i/=:36020'    tan.  9.866564 

T  -  £  a?i  =  634.496          2.802429 
T-  754.496 

Table  A  gives,  for  n  =  8,  d,  =  f  sin.  D  =  f  x  .05814  =  .021802, 
and  d, ,  multiplied  in  succession  by  8,  27,  64,  125,  216,  and  343, 
gives  d*  =  .174,  d3  =  .589,  d*  =  1.395,  d5  =  2.725,  d*  =  4.709,  and 
d,  -  7.478. 

To  find  A  we  have  (§  119)  tan.  A  =  .3  n  sin.  D.  For  small  an- 
gles we  may  put  A  =  .3  n  D.  In  this  example  A  =  2.4  D  —  8°,  and 
the  central  angle  of  the  circular  curve  /  —  2  A  =  56°  40'.  This 
divided  by  2  D  gives  8.5,  as  the  number  of  100  feet  chords  from 
C  to  C'.  * 

123.  Example,  when  T  is  given.    Given  I  -  68°  20',  T  =  764.3, 
and  n  =  5.     Here  xl  =  150,  and  T  —  i  xl  =  689.3. 

689.3  2.838408 

34°  10'    cot.  0.168291 


#  +  iyi  =  1015.5  3.006699 

Comparing  this  approximate  value  of  R  with  values  given  in 
Table  I.,  we  see  that  D  =  2°  50'  might  be  selected  as  a  convenient 
deflection  angle.  We  have  then  R  —  1011.51,  sin.  D  =  sin.  2°  50'  = 
.04943,  y,  =  75  x  .04943  =  3.70725,  and  R  +  iy,  =  1012.44,  to  find 
the  new  T. 

1012.44  3.005369 

i/=34°  10'     tan.  9.831709 

T  -  i  xl  =  687.19  2.837078 

T=  762.19 

"We  next  find  di  =  .6  sin.  D,  and  proceed  as  in  the  preceding 
example. 


112  TRANSITION   CURVES. 

124.  Method  by  Deflection  Angles.     The  transition  curve  can 
also  be  laid  out  by  deflection  angles.     These  angles  (fig.  57)  are 

C 


a' 
Fig.  57. 

aAE,lAE,cAE,  etc.  Denote  them  by  ^ ,  82 ,  53 , 8n .    Now 

cd' 
the  tangent  of  any  one  of  these  angles,  as  83 ,  is  tan.  83  =  -r-r,  = 

y  x* 

-.     If  in  equation  (3),  which  is  y  =  ^5 — ,  we  divide  both  sides 

x  0  o  K  x\ 

y         x 
by  x  we  have  -  =  r-^ — .    This  shows  that  the  tangents  of  the  de- 

X        O  Jit  X\ 

flection  angles  are  to  each  other  as  the  squares  of  the  abscissas. 
Now  if  a  tangent  be  drawn  to  the  curve  at  any  point,  as  c,  the 

tangent  of  the  angle  it  makes  with  A  E  is  by  equation  (2)  -~  = 

x* 
r-^ — .    This  is  exactly  three  times  the  tangent  of  the  deflection 

u  £1  X\ 

angle  just  found  for  the  same  point.  This  relation  being  a  gen- 
eral one,  we  have  at  6V,  tan.  C  A  E  =  £tan.  CFE  or  tan.  8n  = 
%  tan.  A.  All  these  angles  are  ordinarily  so  small  that  the  angles 
themselves  may  be  substituted  for  their  tangents.  It  follows  that 
the  deflection  angles  are  to  each  other  as  the  squares  of  the  ab- 
scissas, and  that  8n  =  $  A.  Taking  A  =  .3  n  Z>,  as  found  above, 

nD        ,  _        8n        D        m, 

we  have  5n  =  £  A  =  — — ,  and  81  =  -r  =  T-T—  .     The  successive  an- 
il) fr       LOn 

gles  to  be  laid  off  from  A  E  with  the  transit  at  A  are  therefore 

81  =  —  — .  82  —  45i ,  53  =  9  5i , 5n  —  n2  81 .     The  annexed 

10  n 

table  gives  the  necessary  data  for  curves  from  60  to  300  feet  in 
length.  D  is  the  deflection  angle  of  the  central  curve  for  100  feet 

chords.    For  any  other  chord  c  multiply  the  values  given  by  — . 


THE   CUBIC    PARABOLA. 


113 


Thus  if  D  were  the  deflection  amgle  for  50  feet  chords,  we  should 

nD  D 

nave  A  =  .to  n  D,  on  =  —— ,  and  $1  =  — -  . 
5  5  n 

TABLE   B. 


n 

*=*nD 

6»-  — 

6l  =  lL 

2 

.6D 

.2D 

JL/J 

3 

.9D 

.3D 

To  ^ 

4 

1.2  D 

.4D 

To-^ 

5 

1.5  D 

.5D 

-±Q   D 

6 

1.8  D 

.6Z) 

•fa  D 

7 

2.1  D 

.7D 

Vo  D 

8 

2.4  D 

.8D 

^-D 

9 

2.7  D 

.9D 

^D 

10 

3.0  D 

1.0  D 

^D 

125.  Example.    Taking  the  data  of  the  example  in  §  122,  we 
have  n  =  8,  D  =  3°  20'  =  200'.     Table  B,  for  n  =  8,  gives  A  =  2.4 
D  =  8°,  8»  =  .8D  =  2°  40',  and  ax  =  ^D  =  2'.5.     Multiplying  by 
the  successive  squares,  4,  9,  16,  etc.,  we  have  5i  =  2 '.5,  82  =  10', 
53  =  22'.5,  54  =  40',  85  =  1°  2  .5,  86  =  1°  30',  S7  =  2°  2'.5. 

To  lay  out  the  circular  curve,  set  the  transit  at  <7,  reverse  from 
A,  and  from  the  line  A  C  thus  produced  turn  off  an  angle,  to  the 
left  or  right  as  the  case  may  require,  equal  to  2  3n .  The  line  of 
sight  will  now  be  tangent  to  the  circular  curve. 

ARTICLE  II. — THE  CUBIC   PARABOLA  APPLIED   TO   AN   EXISTING 
CIRCULAR  TRACK. 

126.  Let  A'PQ  (fig.  58)  be  the  existing  track  of  radius  0  A'  — 
0  P  —  R,  and  tangent  at  A'  to  A'L.     From  a  point  P  on  this 
curve  a  circular  curve  G  C  P  of  radius  0'P=  R',  less  than  R,  is 
drawn,  and  having  the  same  central  angle  as  A'PQ.     It  has, 
therefore,  its  tangent  Gr  M  parallel  to  A'L.     A  B  C  is  a  cubic 
parabola,  running  from  a  point  A  on  the  tangent  of  the  original 
curve  to  a  point  C  on  the  new  circular  curve.     Produce  0'  Gr  to 
//,  and  draw  the  chords  J/Pand  Q-  P.     These  chords  are  on  the 
same  straight  line,  because  the  angle  PGM  is  half  the  central 
angle  at  0  and  the  angle  PAL  is  half  the  equal  central  angle  at 
0  ($  2,  III.).     Now  from  the  properties  of  the  cubic  parabola,  al- 
ready explained  (§  116),  we  know  that  A  E  =  Xi  may  be  taken  as 

9 


114 


TKANSITION   CUKYES. 


»  X  2 

bisected  at  H,  and  that  the  shift  G  H  =  s  —       *     ,  or  putting 

50 

Xi  =  3Qn  (8  119),  and  for  72'   its  value  -  — —  ,  we  have  s  = 

sm.  Z) 

f  n2sin.7)',  and  y  =  E  C  =  4s  =  3w2sin.Z>'.    To  obtain  D'  we 
O 


Fig.  58. 


have  sin.  D'  :  sin.  D  =  R  :  R'.    If  we  put  R'  —  m  R,m  being  any 

sin.  D 
assumed  proper  fraction,  sin.  D  =  —    —  . 

Now  J.'  is  a  fixed  point  on  the  ground,  and  if  we  find  the  dis- 
tance AH  to  the  centre  of  Xi ,  the  points  A  and  E  can  be  found 
by  simply  measuring  $%i  =  15  n  each  way  from  //.  To  fix  the 
point  P,  A'L  and  PL  must  be  found. 

Consider  P  M  and  (7iVto  be  tangent  offsets  to  the  curve  G  C  P 
from  the  tangent  G  M,  and  we  have,  very  closely,  G  M :  G  N  — 


CN 
A  G  :  a  P  =  0  0 ' :  0 '  P  -  R  -  m  R :  m  R  =  1  -  m  :  m . 

q-^-.    Also,  CN=EC-EN=4s-s  =  3s  , 
1  —  m 


-.PM  = 
PM 


THE    CUBIC    PARABOLA.  115 

—  -  .    Substituting  this  value  of  77-^.  in  the  expression  for 
o  (1  —  Til)  C  JM 

/        m  /       m 

G  M,  we  have  GM=  &  N  A/  -      -  =  15  n  A/  -          -. 
V  3  (1  -  w)  r    3  (1  -  m) 


NowAII:GM=00':0'P=l-m:m.  .'.AH= 


15  n  (1  —  m)      /       m  1  —  m 

— AJ  — ;.     Squaring  ,  and  putting  it 

m  r    o  (1  —  m)  m  . 

under  the  radical,  we  have,  after  reduction,  AH  =  15  n  A/ . 

V      3m 

Next,  A'L  :  AH '=  0  P:  0  0'  =  1:1  —  m.    .  •  .A'L  =  - —  -  = 

A/  -     —  .    Squaring  the  denominator  1  —  m,  and  put- 

1  —  m  \      3m 

ting  it  under  the  radical,  and  reducing,  we  have  A'L  =  I5n 

.     Lastly,  PL  =  PM  +  ML  =     ™S     +  s  = 

3  m  (1  -  m)  1  -  m 

s 


1  —  m' 

In  deciding  upon  a  proper  value  for  m,  it  is  obvious  that  R  ' 
should  not  differ  much  from  R.  If  we  make  m  =  .9,  the  change 
would  not  be  too  great.  This  value  also  simplifies  the  formulae 
very  much.  Making  m  =  .9,  we  have 


!ndPL  =  -iQs  =  M 

O 

For  the  central  angle  GrO'C  =  A'  of  the  transition  curve,  we 
have,  as  before  (§  119),  sin.  A'  =  .3wsin.  Z>',  and  for  A  =  A'  OP, 

A'L      50n  |/3      SOnsin.Df  3      n  . 
wehave8m.A  =  —  =  ~^~  -^  -  =pm.Dv3  = 

.3  n  sin.  D'  t/3.  The  central  angle  of  C  P,  the  new  circular  curve, 
is  C  O'P  =  A  —  A'.  In  the  expressions  for  sin.  A'  and  sin.  A  sub- 
stitute the  angles  themselves  for  their  sines,  and  we  have  A'  = 
3n  D'  and  A  =  .3  n  D  '  V  3  and  A-  A'  =  .3  nD'  (  V  3  -  1)  = 
.22  nD',  nearly. 

127.  Table  C  gives  the  values  of  these  expressions,  and  also 
those  of  yi  and  di  for  values  of  n  from  2  to  10.  As  already  shown, 
sin.  D'  =  Y  sin.  D,  or,  more  simply,  D'  —  *£•  D.  D  and  D'  are 
deflection  angles  for  100  feet  chords,  but  it  is  easy  to  modify  the 
expressions  for  other  chords. 


116 


TRANSITION   CURVES. 


TABLE  C. 


n 

*i 

A'H 

A'L 

y\ 

di 

PL 

A' 

A-  A' 

2 

60 

5.77 

57.74 

12  sin.  D' 

§  sin.  D' 

2.5y, 

.6D' 

.44D' 

3 

90 

8.66 

86.60 

27  sin.  D' 

sin.  D' 

2.5T/! 

.9D' 

.66  IX 

4 

120 

11.55 

115.47 

48  sin.  D' 

|  sin.  D' 

2.5  T/J 

1.2D' 

.88D' 

5 

150 

14.43 

144.34 

75  sin.  D' 

§  sin.  D' 

2.5  y, 

1.5  D' 

1.10D' 

6 

180 

17.32 

173.21 

108  sin.  D' 

£  sin.  Z>' 

2.50J 

1.8D' 

1.32D' 

7 

210 

20.21 

202.07 

147  sin.  D' 

?  sin.  D' 

2.50! 

2.1Z>' 

1.54D' 

8 

240 

23.09 

230.94 

192  sin.  D' 

|  sin.  D' 

2.60, 

2  AD' 

1.76D' 

9 

270 

25.98 

259.80 

243  sin.  D' 

£  sin.  D' 

2.5s/! 

2.7  D' 

1.98D' 

10 

300 

28.87 

288.68 

300  sin.  D' 

$s  sin.  D' 

2.5  2/1 

3.0D' 

2.20  D' 

128.  Example.    Given  the  deflection  angle  D  =  3°  of  an  exist- 
ing circular  track  A'P  Q  (fig.  58).     We  have  for  the  deflection  an- 
gle of  the  curve  G  C  P,  D '  =  ^  D  =  3°  20'.     Take  a*  =  150  feet, 
and  we  have  from  Table  C,  for  n  =  5,  A'JT=  14.43,  A'L  =  144.34, 
yl  =  75  sin.  3°  20'  =  75  x  .05814  =  4.36,  d,  =  .6  x  .05814  =  .03488, 
and  P  £  =  10.90.    From  the  known  tangent  point  A'  of  the  existing 
track  A'P  Q,  we  measure  14.43  feet  to  H,  and  from  H  75  feet  each 
way  to  A  and  E.    Then  the  point  P  is  fixed  by  A'L  =  144.34  and 
PL  =  10.90.     The  transition  curve  is  then  put  in  by  offsets  from 
the  tangent  A  E.    Th^se  offsets  are  d^  —  .03488,  dz  =  8  d*  =  .279, 
d3  =  27  di  =  .942,  d4  =  64  d,  =  2.232,  d5  =  yl=  4.36.     The  central 
angle  of  the  short  circular  curve  OP  is  A  —  A'  —  1.1  D'  —  3°  40'. 
As  the  central,  angle  of  this  curve  for  a  chord  of  100  feet  is  2  D\ 
the  chord  (7Pwill  be  the  same  part  of  100  feet  that  1.1  Z)'  is  of 
2  D'  or  55  feet,  and  if  the  work  is  correct,  this  will  be  the  distance 
on  the  ground.     A  further  check  would  be  to  find  the  tangent  at 
C,  and  compute  the  proper  offset  to  P.     In  regard  to  this  check, 
it  should  be  observed  that  the  value  PL  —  2.5^  is  not  exact,  as 
it  depends  upon  the  assumption  that  C  N  :  P  M  =  a  N*  :  &  M \ 
which  is  not  strictly  true.     PL  may  be  computed  accurately  by 
the  formula  PL  =  R-  OK=R-  *J R*  -  A'L*.     The  radical 
under  the  form  \/(R  4-  A'L)  (R  —  A'L)  is  easily  computed  by 
logarithms.     In  the  present  case  we  should  find  PL  —  10.966. 

129.  Length  of  Curve  in  Terms  of  its  Chords. — The  length  of  a 
transition  curve,  as  measured  by  the  sum  of  the  chords  used  in 
laying  it  out,  is  slightly  in  excess  of  the  abscissa  xl .     This  excess 
is  generally  so  small  that  it  may  be  neglected.     When,  however, 
the  curve  is  long,  and  the  deflection  angle  of  the  circular  curve 


THE    CUBIC    PARABOLA. 


117 


large,  a  method  of  calculating  the  excess  may  be  desirable.  Each 
chord  is  the  hypothenuse  of  a  right-angled  triangle,  whose  base  is 
30  feet,  and  perpendicular  the  difference  between  two  successive 
tangent  offsets.  These  offsets  are  di ,  8  di ,  27  di ,  64  dl ,  etc.,  and  the 
successive  differences  or  perpendiculars  are  di ,  7  di ,  19  di ,  37  dl , 
etc.  Let  p  denote  any  one  of  these  perpendiculars,  and  for  the 
corresponding  chord  c  we  have  c  =  -\/302  +  p12.  By  developing 
this  radical,  and  retaining  the  first  two  terms  only  of  the  root,  we 

have  c  =  30  +  |-  ,  nearly.    Substituting  for  p  its  successive  values, 

the  excess  of  the  first  chord  will  be  ^r,  of  the  second  chord, 

49 dS      ,  ,.      ,,  .   ,    361  dS 

._     ,  of  the  third,  — ^7— ,  etc.    For  a  curve  of  n  chords  we 
oU  oU  , 2 

should  have  for  e,  the  total  excess,  e  =  ^  (I2  +  72  +  19*  +  37*  + 
etc.),  the  parenthesis  containing  always  n  terras  of  the  series.  For 
di  substitute  its  value  already  found  di  =  —  —  (§  120),  D  being 
the  deflection  angle  of  the  circular  curve  for  100  feet  chords,  and 
we  have,  after  reducing,  e  =  >l0  s™'* D  (I2  +  72  +  192  +  372  + 

etc.).  If  e  is  computed  by  this  formula  for  D  =  1°,  and  different 
values  of  n.  the  excess  for  any  other  deflec- 
tion angle  Di ,  and  given  n  will  be  obtained, 
very  closely,  by  multiplying  the  value  so 
found  for  D  =  1°  and  the  given  n  by  the 
square  of  the  number  denoting  Z),  in  de- 
grees. The  values  of  e  for  Z>  =  1°,  and 
values  of  n  from  2  to  10  have  been  calcu- 
lated, and  the  results  placed  in  the  annexed 
table,  where  e2  is  the  excess  for  n  =  2,  e9  the 
excess  for  n  =  3,  etc. 

130.  Example.  Given  the  deflection  angle  of  the  circular  curve 
=  3£°  =  J°,  and  n  —  6,  to  find  the  excess  of  the  length  of  the 
transition  curve  measured  by  its  chords  over  x\ .  Here  we  multi- 
ply e6  in  the  table  by  (|)2  =  ±4a,  and  we  have  the  excess  e  =  .01749 
x  \a  =  .21425.  For  n  =  6,  xl  =  180,  so  that  the  length  of  the 
curve  by  chords  is  180.214. 


118 


TRANSITION    CURVES. 


ARTICLE  III. — CURVING  THE  RAILS. 

131.  To  secure  the  greatest  ease  of  motion  on  a  transition  curve, 
it  is  of  importance  that  the  rails  be  properly  curved.  To  do  this 
we  must  have,  as  on  a  circular  curve  (§  28),  the  middle  ordinate 
and  the  ordinates  at  the  quarter  points.  We  there  found  that 
the  ordinates  at  the  quarter  points  were  each  f  m,  m  being  the 
middle  ordinate.  Here  we  shall  find  that  the  ordinate  at  the  first 
quarter  point  is  slightly  less  than  f  m  and  the  ordinate  at  the  sec- 
ond quarter  point  slightly  greater  than  f  m.  This  is  what  might 
be  expected  from  the  gradual  increase  of  the  curvature. 

Let  A  G  B  (fig.  59)  be  a  rail  length  on  any  part  of  a  transition 
curve,  and  CD  its  projection  on  the  axis  of  x.  Let  C  be  distant 


from  the  origin  r  rail  lengths,  and  D  distant  r  +  1  rail  lengths,  r 
being  a  whole  or  fractional  number.  Let  d^ ,  as  above,  denote  the 
tangent  offset  at  the  end  of  the  first  rail  length  from  the  origin. 
Then  the  offset  A  C  =  r3  dl ,  and  the  offset  B  D  =  (r  +  I)3  dl .  The 
middle  ordinate  for  curving  the  rail  will  be  m  =  GF=EF  — 

EG.    Now  EF=i(A  C  +  B  D)  =  (r3  +  r3  +  3r2  +  3r  +  1)  —  = 

(r8  +  f  r2  +  f  r  +  •$•)  ^  and  E  G  —  (r  +  l)3^  =  (r3  +  f  r2  -+-  f  r  +  •}•)  cZv 
Subtracting  and  reducing,  we  have 

m  =  f  (2  r  +  1)  dl . 

In  a  similar  way  the  ordinates  HI  and  KL  at  the  quarter  points 
are  found.  They  are 

H I  =  (-!%  r  +  £{ )  di  =  f  m  —  ^4-  ^ , 
K  L  =  (-£-s  r  _|_  |i)  ^  —  |  m  +  ^.  ^  ^ 

If  the  curve  does  not  begin  at  a  joint,  that  part  of  a  rail  that 
comes  on  the  curve  may  be  curved  by  finding  the  proper  tangent 


COMPOUND    TRANSITION    CURVE.  119 

offset  for  its  length,  and  bending  the  end  from  the  straight  line  a 
distance  equal  to  the  offset.  As  the  tangent  offset  for  a  whole 
rail  is  di,  the  offset  for  a  fraction  will  be  di  multiplied  by  the 
cube  of  the  fraction.  Thus,  if  the  fraction  is  .8  the  offset  would 
be  .512  di .  Except  in  extreme  cases,  this  offset  is  so  small  that 
the  rail  remains  practically  straight. 

If  the  curve  begins  at  a  joint  the  middle  ordinates  for  the  suc- 
cessive rails  will  be  obtained  by  making  r  successively  0,  1,  2,  3, 
etc.  Denoting  these  ordinates  by  Wi,  ma,  ms,  etc.,  we  have  m-i  = 
f  di ,  m-9  =  £  di ,  ms  =  J85  di ,  etc.,  or  ml  =  f  di ,  w2  =  3  ml ,  ms  = 
5  wii,  m4=7wi,  etc.  Taking  three  fourths  of  these  ordinates, 
and  subtracting  and  adding  -fcdi,  we  have  the  quarter  point 
ordinates. 

ARTICLE  IV.— COMPOUND  TRANSITION  CURVE. 

13.2.  Transition  curves  of  this  kind  consist  of  successive  circu- 
lar arcs,  the  deflection  angles  of  which  are  such  that  if  D  is  the 
deflection  angle  of  the  first  arc,  that  of  the  second  is  2  />,  that  of 
the  third  3  Z>,  and  so  on.  The  chords  are  all  of  the  same  length. 
A  curve  of  this  kind  A  B  CD  (fig.  60)  may  be  readily  laid  out  by 
offsets  from  the  tangent  A  /,  measuring  at  the  same  time  the 
successive  chords.  Let  c  represent  the  length  of  each  chord,  n 
their  number,  and  let  D  be  the  deflection  for  the  first  chord,  2  D 
that  for  the  second  chord,  3  D  that  for  the  third  chord,  and  so  on 
to  the  deflection  angle  ot  the  last  chord,  which  will  be  n  D.  Then 
it  is  easily  seen  that  the  angles  Ti  AB,T*B  C,  Tz  CD,  etc.,  will 


Fig.  60. 


be  successively  D,  4  D,  9  D,  16  Z),  etc.,  up  to  n8  D.  Calling  the  re- 
quired offsets  from  the  tangent  A  7,  di,  d2,  d3,  etc.,  and  recollect- 
ing that,  since  these  angles  are  all  small,  we  may  put  sin.  4  D  = 
4  sin.  D,  sin.  9  D  —  9  sin.  Z),  etc.,  we  have  di  =  c  sin.  D,  d^  =  di  + 
4  c  sin.  D  —  dl  +  4  di  =  5  dl ,  ds  =  d*  +  9  c  sin.  D  =  5  dl  +  9  di  = 


120 


TRANSITION   CURVES. 


14  dl ,  etc..  the  successive  offsets  being  formed  by  multiplying  di 
by  the  terms  of  the  series  1,  5,  14,  30,  55,  91,  etc.,  formed  by  the 
successive  additions  of  the  squares  of  the  natural  numbers. 

More  accurate  values  of  the  offsets  may  be  obtained  thus.  From 
the  table  of  natural  sines,  set  down  in  a  column  sin.  7),  sin.  4  Z>, 
sin.  9  D,  etc.,  up  to  sin.  n*  D.  Then  for  dl ,  d% ,  da ,  etc.,  multiply 
successively  by  c  the  first  number  so  set  down,  the  sum  of  the  first 
two  numbers,  the  sum  of  the  first  three  numbers,  and  so  on,  until 
for  dn  multiply  by  c  the  sum  of  the  whole  column. 

The  projections  of  the  chords  A  TI,  B  T2,  C  T3,  etc.,  may  be 
found  thus.  A  1\  -  c  cos.  D,BT^-c  cos.  4  Z),  C  T3  =  ccos.  9D, 
etc.  From  the  table  of  natural  cosines,  set  down  in  a  column 
cos.  Z),  cos.  4  D,  cos.  9  D,  etc.,  up  to  cos.  n*  D.  Denote  by  p^ ,  p? , 
pa,  etc.,  respectively,  the  first  projection,  the  sum  of  the  first  two 
projections,  the  sum  of  the  first  three  projections.  Then  to  obtain 
Pi,p-i,p3,  etc.,  multiply  successively  by  c  the  first  number  in  the 
column,  the  sum  of  the  first  two  numbers,  the  sum  of  the  first 
three  numbers,  and  so  on,  until  for  pn  multiply  by  c  the  sum  of 
the  whole  column. 

133.  We  have  now  to  find  (fig.  61)  A  I  =  T,  when  R  the  radius 
of  the  central  curve  is  given,  or  to  find  R,  when  Tis  given.  In 


both  cases  the  intersection  angle  /  is  supposed  to  be  known,  and 
the  number  n  of  chords  in  the  transition  curve  to  be  assumed. 


134.  Problem.  Given  the  intersection  angle  I  and  the  ra- 
dius 0  C  —  R  or  the  deflection  angle  D'  of  C M,  the  main  or  cen- 
tral curve  (fig.  61),  to  find  the  deflection  angle  D  for  the  first  arc 


COMPOUND  TRANSITION  CURVE.  121 

of  the  transition  curve  A  C,  the  coordinates  A  E  =  a  and  E  C  = 
b  of  the  point  C,  and  the  tangent  A  I. 

Solution.  Let  the  number  of  chords  in  A  C  be  denoted  by  n, 
and  the  length  of  each  chord  by  c.  C  M  is  half  the  central  curve, 
so  that  the  angle  H  0  1  =  |  /.  Run  C  M  back  to  6r,  where  its 
tangent  becomes  parallel  to  A  /,  and  draw  0  G  H  and  C  K.  De- 
note the  deflection  angle  of  the  central  curve  for  a  chord  equal  to 
c  by  D'.  This  deflection  angle  is  either  given  directly,  or  found 
from  that  given  for  a  different  chord.  Then  as  D  is  the  deflection 
angle  of  the  first  chord  on  A  C,  the  deflection  angle  for  the  last 
chord  will  be  n  Z>,  and  for  the  first  on  C  M,  (n  +  1)  D  =  D  ' 


Having  D,  we  have  also  (§  132)  di  ,  d*  ,  d9  ,  etc.  From  the  pre- 
ceding section,  we  have 

a  =  A  E  =  c  (cos.  D  +  cos.  4  D  +  cos.  9  D  +  •  •  •  cos.  n*D) 
=  n  c,  nearly. 

&  =  E  C  =  c  (sin.  D  4-  sin.  4  D  +  sin.  9  D  +  •  •  •  sin.  n*D) 

=  di  (1  +  4  +  9  +  •  •  •  •  ft2),  nearly 

To  find  T  we  have  T=  A  H  +  HI.  Now  A  H=  A  E  -  HE  - 
a  —  R  sin.  COG.  The  angle  C  0  &  is  the  sum  of  the.central  an- 
gles of  the  seyeral  arcs  of  A  C.  The  central  angle  of  the  first  arc 
is  twice  its  deflection  angle,  or  2  Z>,  that  of  the  second  arc  is  2  x 
2  Z),  of  the  third  2x37),  etc.  Denote  the  sum  of  these  angles  by 
o,  and  we  have 

a  =  2  Z>  (1  +  2  +  3  -f  •  •  •  •  n)  =  n  (n  +  1)  D. 
Therefore  AH  =  AE  —  HE  =  a  —  R  sin.  a. 

Next,  HI  =  0  H  tan.  HOI=(EC  +  OK)  tan.  \  7,  or  HI- 
(l)  +  R  cos.  o)  tan.  1  1.  Substituting  these  values  of  A  H  and 
HI,  we  have 

B^~          T=  a  —  R  sin.  a  +  (&  +  R  cos.  a)  tan.  |  J. 

An  approximate  formula  for  T,  generally  accurate  enough  in 
practice,  may  be  found  thus.  Consider  HE  to  be  equal  in  length 
to  the  arc  (r  C  and  find  the  length  of  &  C  in  chords  of  length  c 
by  dividing  half  its  central  angle  or  4-  a  by  its  deflection  angle 


D1  =  (n  +  1)  D.    Hence  HE  =  _     =  ±nc  and  A  j/  = 

(n  4-  1)  7) 


Also,  7T/=  0  J^tan.  i/  = 


TRANSITION    CURVES. 


(R  +  G  II)  tan.  -£  /.  Omit  G II  as  small  relatively  to  R,  and  we 
have  P.  I—  R  tan.  4-  Z  Substituting  these  values  of  J.  H  and 
# I  in  the  formula  T=  A  H  +  HI,  we  have 

T  =  i  7i  c  +  R  tan.  £  /,  nearly. 


135.  Example.  Given  1  =  42°,  the  deflection  angle  of  the  cen- 
tral curve  =  2°  for  100  feet  chords,  n  =  5,  and  c  =  30,  to  find  the 
deflection  angle  D  of  the  first  arc  of  the  transition  curve  A  C 
(fig.  61),  the  coordinates  a  and  b  of  the  point  C,  and  the  tangent 
A  I  =  T. 

Here  the  deflection  angle  of  the  central  curve  for  30  feet  chords 


'  =  7^  x  2°  =  36'  and  D  = 


n  +  1       6 

csin.Z)  =  30  x  .001745  =:  .05235.  Computing  by  the  exact  for- 
mulae we  find  a  =  149.956,  b  =  2.879,  and  T  =  625.24.  By  the 
approximate  formulae,  we  find  a  =  150,  b  =  2.879,  and  T  =  624.85. 

136.  Problem.  Given  the  intersection  angle  7,  and  the  tan- 
gent A  1=  T,  to  find  the  radius  0  C  =  R  of  the  central  curve 
CM  (fig.  61). 


Solution.    From  the  preceding  section  we  have  T= 
R  tan.  %  I,  nearly. 

t.  |  J,  nearly. 


This  approximate  value  of  R  may  now  be  substituted  in  the  exact 
formula  for  Tin  the  preceding  section,  and  if  the  value  of  Tthus 
found  does  not  change  the  tangent  point  too  much,  this  value  of 


COMPOUND   TRANSITION    CURVE.  123 

R  may  stand,  and  D',  D,  and  the  other  requisite  data  be  com- 
puted. 

The  principal  inaccuracy  in  the  formula  for  R  is  due  to  drop- 
ping Gr  H  in  the  expression  for  H  7,  above.  If  we  retain  O  77,  we 
should  find 

R-  (r-inc)cot.i/-  Q  H. 


To  get  a  more  accurate  value  of  R,  subtract  G  H,  which  may  be 
computed  by  the  formula  G  U=E  C  —  K  G  =  b  —  R(l  —  cos.  a). 

Generally,  however,  the  approximate  value  of  R  would  be  used 
only  for  finding  a  convenient  deflection  angle  for  the  central 
curve  —  that  is,  one  not  involving  seconds.  A  new  value  of  R 
would  result,  and  a  new  value  of  T  would  have  to  be  computed. 

137.  To  run  the  central  curve  C  M,  we  must  be  able  to  fix  the 
common  tangent  C  F.  This  may  be  readily  done  if  we  find  the 
distance  F  E.  Now  in  the  triangle  C  F  E  the  angle  C  F  E  has 
its  sides  perpendicular  to  those  of  the  angle  C  0  G,  and  is,  there- 
fore, =  a  =  n  (n  +  1)  D. 

gy  .  •  .  F'E  =  b  cot.  a  =  b  cot.  n(n  +  l)  D. 

The  central  angle  of  the  central  curve  will  be  2@OM—2a  = 
I  —  2  n  (n  4-  1)  7),  and  the  number  of  chords  will  be  found  in  the 
usual  way  by  dividing  the  central  angle  by  twice  the  deflection 
angle  used  in  laying  out  the  curve. 

137.  Remark.  There  are  certain  advantages  in  beginning  a  tran- 
sition curve  at  a  joint.  The  ends  of  each  rail  would  then  be  defi- 
nitely fixed  by  the  offsets,  and  the  rails  could  be  more  satisfactorily 
curved.  It  would  be  easier  to  maintain  the  track  in  its  proper 
position,  if  the  trackmen  knew  that  the  tangent  point  was  at  a 
joint,  and  when  the  rails  were  renewed,  the  new  rails  would  be  more 
likely  to  be  properly  curved,  and  placed  in  their  true  position. 


124  LEVELLING. 

CHAPTEE  IV. 

LEVELLING. 
ARTICLE  I. — HEIGHTS  AND  SLOPE  STAKES. 

138.  THE  Level  is  an  instrument  consisting  essentially  of  a  tele- 
scope, supported  on  a  tripod  of  convenient  height,  and  capable  of 
being  so  adjusted  that  its  line  of  sight  shall  be  horizontal,  and 
that  the  telescope  itself  may  be  turned  in  any  direction  on  a  ver- 
tical axis.     The  instrument  when  so  adjusted  is  said  to  be  set. 

The  line  of  sight,  being  a  line  of  indefinite  length,  maybe  made 
to  describe  a  horizontal  plane  of  indefinite  extent,  called  the  plane 
of  the  level. 

The  levelling  rod  is  used  for  measuring  the  vertical  distance  of 
any  point,  on  which  it  may  be  placed,  below  the  plane  ol  the  level. 
This  distance  is  called  the  sight  on  that  point. 

139.  Problem.     To  find  the  difference  of  level  of  two  points, 
as  A  and  B  (fig.  62). 

Solution.  Set  the  level  between  the  two  points,*  and  take 
sights  on  both  points.  Subtract  the  less  of  these  sights  from  the 
greater,  and  the  difference  will  be  the  difference  of  level  required. 
For  if  F  P  represent  the  plane  of  the  level,  and  A  G  be  drawn 
through  A  parallel  to  F  P,  A  F  will  be  the  sight  on  A,  and  B  P 
the  sight  on  B.  Then  the  required  difference  of  level  B  Q  = 
BP-PG=  BP-AF. 

If  the  distance  between  the  points,  or  the  nature  of  the  ground, 
makes  it  necessary  to  set  the  level  more  than  once,  set  down  all 
the  backward  sights  in  one  column  and  all  the  forward  sights  in 
another.  Add  up  these  columns,  and  take  the  less  of  the  two 
sums  from  the  greater,  and  the  difference  will  be  the  difference  of 
level  required.  Thus,  to  find  the  difference  of  level  between  A 
and  D  (fig.  62),  the  level  is  first  set  between  A  and  B,  and  sights 


*  The  level  should  be  placed  midway  between  the  .two  points,  when  prac- 
ticable, in  order  to  neutralize  the  effect  of  inaccuracy  in  the  adjustment  of 
the  instrument,  and  for  the  reason  given  in  §  148. 


HEIGHTS   AND   SLOPE   STAKES. 


125 


Ed      O 


are  taken  on  A  and  B ;  the  level  is  then  set  between  B  and  C,  and 

sights  are  taken  on  B  and  C ;  lastly, 

the  level  is  set  between  C  and  Z),  and 

sights  are  taken  on  C  and  D.     Then 

the  difference  of  level  between  A  and 

DisED  =  (BP+KC+  0  D) - 

(AF  +  B  I  +  NC).     For  J£  D  = 

H  C  -  L  C  =  II M  +  M  C  -  L  C. 

But  H  M  =  B  G  =  B  P-  A  F,  M  C 

=  KC-BI,   and   LC=NC- 

0  D.     Substituting  these  values,  we 

have  ED  =  B  P-  AF  +  KC  - 

BI-NC  +  OD=  (BP  +  KG  + 

OD)-(AF  +  BI  +  NC). 

140.  It  is  often  convenient  to  refer 
all  heights  to  an  imaginary  level 
plane  called  the  datum  plane.  This 
plane  may  be  assumed  at  starting  to 
pass  through,  or  at  some  fixed  dis- 
tance above  or  below,  any  permanent 
object,  called  a  bench-mark,  or  simply 
a  bench.  It  is  most  convenient,  in 
order  to  avoid  minus  heights,  to  as- 
sume the  datum  plane  at  such  a  dis- 
tance below  the  bench-mark,  that  it 
will  pass  below  all  the  points  on  the 
line  to  be  levelled.  Thus  if  A  B  (fig. 
63)  were  part  of  the  line  to  be  lev- 
elled, and  if  A  were  the  starting 
point,  we  should  assume  the  datum 
plane  CD  at  such  a  distance  below 
some  permanent  object  near  A,  as 
would  make  it  pass  below  all  the 
points  on  the  line.  If,  for  instance, 
we  had  reason  to  believe  that  no 
point  on  this  line  was  more  than  15 
or  20  feet  below  A,  we  might  safely 
assume  CD  to  be  25  feet  below  the 
bench  near  A,  in  which  case  all  the  distances  from  the  line  to  the 
datum  plane  would  be  positive.  Lines  before  being  levelled  are 


126 


LEVELLING. 


usually  divided  into  regular  stations,  the  height  of  each  of  which 
above  the  datum  plane  is  required. 


141.  Problem.  To  find  the 
heights  above  a  datum  plane  of  the 
several  stations  on  a  given  line. 

Solution.  Let  A  B  (fig.  63)  repre- 
sent a  portion  of  the  line,  divided 
into  regular  stations,  marked  0,  1,  2, 
3,  4,  5,  &c.,  and  let  C  D  represent  the 
datum  plane,  assumed  to  be  25  feet 
below  a  bench-mark  near  A.  Sup- 
pose the  level  to  be  set  first  between 
stations  2  and  3,  and  a  sight  upon 
the  bench-mark  to  be  taken,  and 
found  to  be  3.125.  Now  as  this  sight 
shows  that  the  plane  of  the  level  E  F 
is  3.125  feet  above  the  bench-mark, 
and  as  the  datum  plane  is  25  feet  be- 
low this  mark,  we  shall  find  the 
height  of  the  plane  of  the  level  above 
the  datum  plane  by  adding  these 
heights,  which  gives  for  the  height 
of  E  F,  25  +  3.125  =  28.125  feet.  This 
height  may  for  brevity's  sake  be 
called  the  height  of  the  instrument, 
meaning  by  this  the  height  of  the 
line  of  sight  of  the  instrument. 

If  now  a  sight  be  taken  on  station 
0,  we  shall  obtain  the  height  of  this 
station  above  the  datum  plane,  by 
subtracting  this  sight  from  the  height 
of  the  instrument ;  for  the  height  of 
this  station  is  0  C  and  QG=EC  — 
E  0.  Thus  if  E  0  =  3.413,  0  C  = 
28.125  -  3.413  =  24.712.  In  like 
manner,  the  heights  of  stations  1,  2, 
3,  4,  and  5  may  be  found,  by  taking 
sights  on  them  in  succession,  and 
subtracting  these  sights  from  the 


HEIGHTS    AND    SLOPE    STAKES.  127 

height  of  the  instrument.     Suppose  these  sights  to  be  respective- 
ly 3.102,  3.827,  4.816,  6.952,  and  9.016,  and  we  have 

height  of  station  0  =  28.125  -  3.413  =  24.712, 

"      "        1  =  28.125  -  3.102  =  25.023, 

"      "        2  =  28.125-3.827  =  24.298, 

"        "      "        3  =  28.125-4.816  =  23.309, 

"        "      "        4  =  28.125-6.952  =  21.173, 

"      "        5  =  28.125  -  9.016  =  19.109. 

Next,  set  the  level  between  stations  7  and  8,  and,  as  the  height 
of  station  5  is  known,  take  a  sight  upon  this  point.  This  sight, 
being  added  to  the  height  of  station  5,  will  give  the  height  of  the 
instrument  in  its  new  position ;  for  GIC=@5  +  5K.  Suppose 
this  sight  to  be  #5  =  2.740,  and  we  have  GK=  19.109  +  2.740  = 
21.849.  A  point  like  station  5,  which  is  used  to  get  the  height  of 
the  instrument  after  resetting,  is  called  a  turning  point.  The 
height  of  the  instrument  being  found,  sights  are  taken  on  stations 
6,  7,  8,  9, 10,  and  the  heights  of  these  stations  found  by  subtracting 
these  sights  from  the  height  of  the  instrument.  Suppose  these 
sights  to  be  respectively  3.311,  4.027,  3.824,  2.516,  and  0.314,  and 
we  have 

height  of  station  6  =  21.849  -  3.311  =  18.538, 

"      "        7  =  21.849-4.027  =  17.822, 

"      "        8  =  21.849  -  3.824  =  18.025, 

"        "      "        9  =  21.849  -  2.516  =  19.333, 

"        "      "      10  =  21.849-0.314  =  21.535. 

The  instrument  is  now  again  carried  forward  and  reset,  station 
10  is  used  as  a  turning  point  to  find  the  height  of  the  instrument, 
and  everything  proceeds  as  before. 

At  convenient  distances  along  the  line,  permanent  objects  are 
selected,  and  their  heights  obtained  and  preserved,  to  be  used  as 
starting  points  in  any  further  operations.  These  are  also  called 
benches.  Let  us  suppose,  that  a  bench  has  been  thus  selected  near 
station  9,  and  that  the  sight  upon  it  from  the  instrument,  when 
set  between  stations  7  and  8,  is  2.635.  Then  the  height  of  this 
bench  will  be  21.849  —  2.635  =  19.214. 

142.  From  what  has  been  shown  above,  it  appears  that  the  first 
thing  to  be  done,  after  setting  the  level,  is  to  take  a  sight  upon 
some  point  of  known  height,  and  that  this  sight  is  always  to  be 
added  to  the  known  height,  in  order  to  get  the  height  of  the  in- 


128 


LEVELLING. 


strument.  This  first  sight  may  therefore  be  called  a  plus  sight. 
The  next  thing  to  be  done  is  to  take  sights  on  those  points  whose 
heights  are  required,  and  to  subtract  these  sights  from  the  height 
of  the  instrument,  in  order  to  get  the  required  heights.  These 
last  sights  may  therefore  be  called  minus  sights. 

143.  The  field  notes  are  kept  in  the  following  form :  The  first 
column  in  the  table  contains  the  stations,  and  also  the  benches 
marked  B.,  and  the  turning  points  marked  t.  p.,  except  when  co- 
incident with  a  station.  The  second  column  contains  the  plus 
sights  ;  the  third  column  shows  the  height  of  the  instrument ;  the 
fourth  contains  the  minus  sights ;  and  the  fifth  contains  the 
heights  of  the  points  in  the  first  column.  The  height  of  the  bench 


Station. 

+  s. 

H.I. 

-S. 

H. 

B. 

3.125 

25.000 

0 

28.125 

3.413 

24.712 

1 

3.102 

25.023 

2 

3.827 

24.298 

3 

4.816 

23.309 

4 

6.952 

21.173 

5 

2.740 

9.016 

19.109 

6 

21.849 

3.311 

18.538 

7 

4.027 

17.822 

8 

3.824 

18.025 

9 

2.516 

19.333 

B. 

2.635 

19.214 

10 

0.314 

21.535 

is  set  down  as  assumed  above,  namely,  25  feet ;  the  first  plus  sight 
is  set  opposite  B.,  on  which  point  it  was  taken,  and,  being  added 
to  the  height  in  the  same  line,  gives  the  height  of  the  instrument, 
which  is  set  opposite  0;  the  minus  sights  are  set  opposite  the 
points  on  which  they  are  taken,  and.  being  subtracted  from  the 
height  of  the  instrument,  give  the  heights  of  these  points,  as  set 
down  in  the  fifth  column.  The  minus  sights  are  subtracted  from 
the  same  height  of  the  instrument,  as  far  as  the  turning  point  at 
station  5,  inclusive.  The  plus  sight  on  station  5  is  set  opposite 
this  station,  and  a  new  height  obtained  for  the  instrument  by  add- 
ing the  plus  sight  to  the  height  of  the  turning  point.  This  new 
height  of  the  instrument  is  set  opposite  station  6,  where  the  minus 
sights  to  be  subtracted  from  it  commence.  These  sights  are  again 
set  opposite  the  points  on  which  they  were  taken,  and,  being  sub- 


HEIGHTS   AND   SLOPE   STAKES. 


129 


tracted  from  the  new  height  of  the  instrument,  give  the  heights 
in  the  last  column. 

144.  Problem.  To  set  slope  stakes  for  excavations  and  em- 
bankments. 

Solution.  Let  A  B  HK  C  (fig.  64)  be  a  cross-section  of  a  pro- 
posed excavation,  and  let  the  centre  cut  A  M  =  c,  and  the  width 
of  the  road-bed  HK=  b.  The  slope  of  the  sides  B  H  or  C  K  is 
usually  given  by  the  ratio  of  the  base  KN  to  the  height  EN. 

Fig.  64. 


Suppose,  in  the  present  case,  that  K N  :  EN  —  3  : 2,  and  we  have 
the  slope  =  f .  Then  if  the  ground  were  level,  as  D  A  E,  it  is  evi- 
dent that  the  distance  from  the  centre  A  to  the  slope  stakes  at  D 
and  J£  would  be  A  D  =  A  E  =  M  K  +  KN=%b  +  f  c.  But  as 
the  ground  rises  from  A  to  C  through  a  height  C  O  =  g,  the  slope 
stake  must  be  set  farther  out  a  distance  E  G  =  f  g ;  and  as  the 
ground  falls  from  A  to  B  through  a  height  B  F  =  g,  the  slope 
stake  must  be  set  farther  in  a  distance  D  F  =  f  g. 

To  find  B  and  (7,  set  the  level,  if  possible,  in  a  convenient  posi- 
tion for  sighting  on  the  points  A,  B,  and  C.  From  the  known 
cut  at  the  centre  find  the  value  ofAE=$b  +  %c.  Estimate  by 
the  eye  the  rise  from  the  centre  to  where  the  slope  stake  is  to  be 
set,  and  take  this  as  the  probable  value  of  g.  Tcr  A  E  add  f  g,  as 
thus  estimated,  and  measure  from  the  centre  a  distance  out,  equal 
to  the  sum.  Obtain  now  by  the  level  the  rise  from  the  centre  to 
this  point,  and  if  it  agrees  with  the  estimated  rise,  the  distance  out 
is  correct.  But  if  the  estimated  rise  prove  too  great  or  too  small, 
assume  a  new  value  for  g,  measure  a  corresponding  distance  out, 
and  test  the  accuracy  of  the  estimate  by  the  level,  as  before. 
These  trials  must  be  continued,  until  the  estimated  rise  agrees 
sufficiently  well  with  the  rise  found  by  the  level  at  the  correspond- 
ing distance  out.  The  distance  out  will  then  be^fc  +  fc  +  f*?. 
10 


130 


LEVELLING. 


The  same  course  is  to  be  pursued,  when  the  ground  falls  from  the 
centre,  as  at  B ;  but  as  g  here  becomes  minus,  the  distance  out, 
when  the  true  value  of  g  is  found,  will  be  A  F  =  A  D  —  D  F  =. 
iZ»  +  £c-f<7. 

For  embankment,  the  process  of  setting  slope  stakes  is  the  same 
as  for  excavation,  except  that  a  rise  in  the  ground  from  the  centre 
on  embankments  corresponds  to  a  fall  on  excavations,  and  vice 
versa.  This  will  be  evident  by  inverting  figure  64,  which  will  then 
represent  an  embankment.  What  was  before  a  fall  to  B,  becomes 
now  a  rise,  and  what  was  before  a  rise  to  C,  becomes  now  a  fall. 

When  the  section  is  partly  in  excavation  and  partly  in  embank- 
ment, the  method  above  applies  directly  only  to  the  side  which  is 
in  excavation  at  the  same  time  that  the  centre  of  the  road-bed  is  in 
excavation,  or  in  embankment  at  the  same  time  that  the  centre  is 
in  embankment.  On  the  opposite  side,  however,  it  is  only  neces- 
sary to  make  c  in  the  expressions  above  minus,  because  its  effect 
here  is  to  diminish  the  distance  out.  The  formula  for  this  dis- 
tance out  will,  therefore,  become  £&  —  f  c  +  f  </. 

In  these  formulae  the  ratio  of  the  base  to  the  height  of  a  slope, 
as  KN :  E  N,  has  been  taken  as  f ,  the  ordinary  ratio  in  earth. 
This  ratio  will,  of  course,  differ  in  different  materials,  and  may  in 
general  be  denoted  by  5.  By  substituting  s  for  f  in  the  preceding 
formulae  they  apply  to  all  slopes. 

The  following  process  is  often  of  advantage  in  setting  slope  stakes. 
Figure  65  represents  the  operation  at  three  successive  stations  : 


Fig.  65. 


Let  C  C  C  represent  the  datum  plane, 
"    B  C  =  height  of  instrument  =  H9 


EARTH'S  CURVATURE  AND  REFRACTION.  131 

Let  CD  —  height  of  road-bed  —  h, 
"    A  B  —  sight  on  the  ground  at  the  supposed 

place  of  side-stake  =  S, 

"    A  D  =  the  side  cut  (minus  cuts  are  fills)  =  c'  ; 
then  in  all  three  of  the  cases  represented 


orc'=H-h-S. 

Having  thus  the  side-cut  or  fill  at  the  supposed  place  for  a 
slope  stake,  we  have  for  the  distance  out  (slope  1.5  to  1)  d  = 
i&  +  fc'. 

For  the  same  setting  of  the  instrument  IT—  h  is  constant  for 
any  one  cross-section,  and  varies  with  h  from  one  station  to  an- 
other. 

It  is  obvious  that  the  cut  or  fill  at  any  point  between  the  side 
stakes  can  be  obtained  in  the  same  manner. 

ARTICLE  II.  —  CORRECTION   FOR   THE    EARTH'S    CURVATURE  AND 
FOR  REFRACTION. 

145.  LET  A  C  (fig.  66)  represent  a  portion  of  the  earth's  surface. 
Then,  if  a  level  be  set  at  J.,  the  line  of  sight  of  the  level  will  be 
the  tangent  A  D,  while  the  true  level  will  be  A  C.    The  difference 
,D  C  between  the  line  of  sight  and  the  true  level  is  the  correction 
for  the  earth's  curvature  for  the  distance  A  D. 

146.  A  correction  in  the  opposite  direction  arises  from  refrac- 
tion.   Refraction  is  the  change  of  direction  which  light  undergoes 
in  passing  from  one  medium  into  another  of  different  density.  As 
the  atmosphere  increases  in  density  the  nearer  it  lies  to  the  earth's 
surface,  light,  passing  from  a  point  B  to  a  lower  point  A,  enters 
continually  air  of  greater  and  greater  density,  and  its  path  is  in 
consequence  a  curve  concave  towards  the  earth.    Near  the  earth's 
surface  this  path  may  betaken  as  the  arc  of  a  circle  whose  radius 
is  seven  times  the  radius  of  the  earth.*    Now  a  level  at  A,  having 
its  line  of  sight  in  the  direction  A  Z>,  tangent  to  the  curve  A  B,  is 
in  the  proper  position  to  receive  the  light  from  an  object  at  B  ;  so 

*  Peirce's  Spherical  Astronomy,  Chap.  X.,  §  125.  It  should  be  observed, 
however,  that  the  effect  of  refraction  is  very  uncertain,  varying  with  the 
state  of  the  atmosphere.  Sometimes  the  path  of  a  ray  is  even  made  convex 
towards  the  earth,  and  sometimes  the  rays  are  refracted  horizontally  as 
well  as  vertically. 


132 


LEVELLING. 


that  this  object  appears  to  the  observer  to  be  at  D.  The  effect  of 
refraction,  therefore,  is  to  make  an  object  appear  higher  than  its 
true  position.  Then,  since  the  correction  for  the  earth's  curvature 
D  C  and  the  correction  for  refraction  D  B  are  in  opposite  direc- 
tions, the  correction  for  both  will  be  B  C  —  D  C  —  D  B.  This 
correction  must  be  added  to  the  height  of  any  object  as  deter- 
mined by  the  level. 

147.  Problem.  Given  the  distance  A  D  =  D  (fig.  66),  the 
radius  of  the  earth  A  E  —  R,  and  the  radius  of  the  arc  of  re- 
fracted light  =  7  R,  to  find  the  correction  B  C  =  d  for  the  earth's 
curvature  and  for  refraction. 


Fig.  66. 


Solution.  To  find  the  correction  for  the  earth's  curvature  D  C, 
we  have,  by  Geometry,  D  C(D  C  +  2EC)  =  A  Z>2,  or  D  C(D  C  +' 
2  R)  =  Z)2.  But  as  D  C  is  always  very  small  compared  with  the 
diameter  of  the  earth,  it  may  be  dropped  from  the  parenthesis, 

and  we  have  D  C  x  2  R  =  D\  or  D  C  =  ~.     The  correction 

&  £i 

for  refraction  D  B  may  be  found  by  the  method  just  used  for 
finding  D  (7,  merely  changing  R  into  7  R.    Hence  D  B  = 


We  have  then  d  =  BC  =  DC- 


d  = 


Z>2 
2R' 


D* 
UR 


UR' 


,  or 


1R 


By  this  formula  Tab.  VIII.  is  calculated,  taking  R  —  20,911,790 
ft.,  as  given  by  Bowditch.    The  necessity  for  this  correction  may 


VERTICAL   CURVES. 


133 


be  avoided,  whenever  it  is  possible  to  set  the  level  midway  between 
the  points  whose  height  is  required.  In  this  case,  as  the  distance 
on  each  side  of  the  level  is  the  same,  the  corrections  will  be  equal, 
and  will  destroy  each  other. 

ARTICLE  III.— VERTICAL  CURVES. 

148.  Vertical  curves  are  used  to  round  off  the  angles  formed  by 
the  meeting  of  two  grades.     Let  A  C  and  CB  (fig.  67)  be  two 
grades  meeting  at  C.     These  grades  are  supposed  to  be  given  by 
the  rise  per  station  in  going  in  some  particular  direction.     Thus, 
starting  from  A,  the  grades  of  A  C  and  C B  may  be  denoted  re- 
spectively by  g  and  g' ;  that  is,  g  denotes  what  is  added  to  the 
height  at  every  station  on  A  (7,  and  g'  denotes  what  is  added  to 
the  height  at  every  station  on  C  B ;  but  since  C  B  is  a  descending 
grade,  the  quantity  added  is  a  minus  quantity,  and  g1  will  there- 
fore be  negative.    The  parabola  furnishes  a  very  simple  method 
of  putting  in  a  vertical  curve. 

149.  Problem.     Given  the  grade  g  of  A  C  (fig.  £7),  the  grade 
g'  of  C  B,  and  the  number  of  stations  n  on  each  side  of  C  to  the 
tangent  points  A  and  B,  to  unite  these  points  by  a  parabolic  verti- 
cal curve. 


Fig.  67. 


Solution.  Let  A  EB  be  the  required  parabola.  Through  B 
and  C  draw  the  vertical  lines  F  K  and  C  H,  and  produce  A  C  to 
meet  F  K  in  F.  Through  A  draw  the  horizontal  line  A  K,  and 
join  A  B,  cutting  C  H  in  D.  Then,  since  the  distance  from  C  to 
A  and  B  is  measured  horizontally,  we  have  A  H—  H  K,  and  con- 
sequently A  D  —  D  B.  The  vertical  line  C  D  is,  therefore,  a  di- 
ameter of  the  parabola  (g  100,  I.),  and  the  distances  of  the  curve 
in  a  vertical  direction  from  the  stations  on  the  tangent  A  F  are 


134 


LEVELLING. 


to  each  other  as  the  squares  of  the  number  of  stations  from  A 
(§  100,  II.).  Thus,  if  a  represent  this  distance  at  the  first  station 
from  A,  the  distance  at  the  second  station  would  be  4  a,  at  the 
third  station  9  a,  and  at  B,  which  is  2  n  stations  from  A,  it  would 

-im  T> 

be  4n2a;  that  is,  F  B  —  4n'2a,  or  a  =  —r~^ .    To  find  a,  it  will 

4  n 

then  be  necessary  to  find  F  B  first.  Through  C  draw  the  hori- 
zontal line  C  6r,  and  we  have,  from  the  equal  triangles  C  F  O  and 
A  C II,  FO  —  CH.  But  C II is  the  rise  of  the  first  grade  g  in  the 
n  stations  from  A  to  (7;  that  is,  C II  =  ng,  or  F  O  =  ng.  OB 
is  also  the  rise  of  the  second  grade  g'  in  n  stations,  but  since  g'  is 
negative  (§  148),  we  must  put  OB—  —  ng'.  Therefore,  F  B  = 
F  O  +  O  B  =  ng  —  ng'.  Substituting  this  value  of  F  B  in  the 

n  a  —  n  a' 
equation  for  a,  we  have  a  —     y      g      ,  or 

*•  •***£ 

The  value  of  a  being  thus  determined,  all  the  distances  of  the 
curve  from  the  tangent  A  F,  viz.  a,  4  a,  9  a,  16  a,  &c.,  are  known. 
Now  if  Tand  T'  be  the  first  and  second  stations  on  the  tangent, 
and  vertical  lines  TPand  T'P'  be  drawn  to  the  horizontal  line 
AK,  the  height  T  P  of  the  first  station  above  A  will  be  g,  the 
height  T'P'  of  the  second  station  above  A  will  be  2g,  and  in  like 
manner  for  succeeding  stations  we  should  find  the  heights  3g,4g, 
&c.  As  we  have  already  found  TM=a,  T'M'  =  4a,  &c.,  we 
shall  have  for  the  heights  of  the  curve  above  the  level  of  A,  M P  = 
TP—  TM=g  —  a,  M'P'  —  T'P'  —  T'M'  =  2g  —  4a,  and  in 
like  manner  for  the  succeeding  heights  Sg  —  9 a,  4g  —  16 a,  &c. 
Then  to  find  the  grades  for  the  curve  at  the  successive  stations 
from  A,  that  is,  the  rise  of  each  height  over  the  preceding  height, 
we  must  subtract  each  height  from  the  next  following  height, 
thus :  (g  —  a)  —  0  =  g  —  a,  (2g  —  4 a)  —  (g  —  a)  =  g  —  3 a,  (3g  — 


The  successive  grades  for  the  vertical  curve  are,  therefore, 
g  —  a,  g  —  3  a,  g  —  5  a,  g  —  7  a,  &c. 


In  finding  these  grades,  strict  regard  must  be  paid  to  the  algebraic 
signs.  The  results  are  then  general  ;  though  the  figure  represents 
but  one  of  the  six  cases  that  may  arise  from  various  combinations 


VERTICAL   CURVES.  135 

of  ascending  and  descending  grades.  If  proper  figures  were  drawn 
to  represent  the  remaining  cases,  the  above  solution,  with  due  at- 
cention  to  the  signs,  would  apply  to  them  all,  and  lead  to  precisely 
the  same  formulae. 

150.  Examples.    Let  the  number  of  stations  on  each  side  of  G 
be  3,  and  let  AC  ascend  .9  per  station,,and  C B  descend  .6  per 

station.    Here  n  =  3,  g  =  .9,  and  g'  =  —  .6.     Then,  a  —  9—^-  = 

.9      (6)  _  1^?  _  tl25}  and  the  grades  from  A  to  B  will  be 
4  x  o  \& 

g  —  a  =  .9  —  .125  =  .775, 
g  -  3  a  =  .9  -  .375  =  .525, 
g-  5a=  .9-  .625  =  .275, 
g-  7a^.9-  .875  =  .025, 
g  -  9  a  =  .9  -  1.125  =  -  .225, 
g  —  11  a  —  .9  —  1.375  =  —  .475. 

As  a  second  example,  let  the  first  of  two  grades  descend  .8  per 
station,  and  the  second  ascend  .4  per  station,  and  assume  two  sta- 
tions on  each  side  of  C  as  the  extent  of  the  curve.  Here  g  =  —  .8, 

g'  =  .4,  and  n  =  2.  Then  a  =  ^ — -^-  —  — jr-  —  —  .15,  and 
the  four  grades  required  will  be 

g-a  =  -  .8  -  (-  .15)  =  -  .8  +  .15  =  -  .65, 
g  _  3  a  =  -  .8  -  (-  .45)  =  -  .8  +  .45  =  -  .35, 
0_5a=_.8-  (-  .75)  =  _  .8  +  .75  =  -  .05, 
g  -  la  =  -  .8  -  (-  1.05)  =  -  .8  +  1.05  =  +  .25. 

It  will  be  seen,  that,  after  finding  the  first  grade,  the  remaining 
grades  may  be  found  by  the  continual  subtraction  of  2  a.  Thus,  in 
the  first  example,  each  grade  after  the  first  is  .25  less  than  the 
preceding  grade,  and  in  the  second  example,  a  being  here  nega- 
tive, each  grade  after  the  first  is  .3  greater  than  the  preceding 
grade. 

151.  The  grades  calculated  for  the  whole  stations,  as  in  the  fore- 
going examples,  are  sufficient  for  all  purposes  except  for  laying 
the  track.    The  grade  stakes  being  then  usually  only  20  feet  apart, 
it  will  be  necessary  to  ascertain  the  proper  grades  on  a  vertical 
curve  for  these  sub-stations.     To  do  this,  nothing  more  is  neces- 
sary than  to  let  g  and  g'  represent  the  given  grades  for  a  sub-sta- 
tion of  20  feet,  and  n  the  number  of  sub-stations  on  each  side  of 


136  LEVELLING. 

the  intersection,  and  to  apply  the  preceding  formulae.  In  the  last 
example,  for  instance,  the  first  grade  descends  .8  per  station,  or  .16 
every  20  feet,  the  second  grade  ascends  .4  per  station,  or  .08  every 
20  feet,  and  the  number  of  sub-stations  in  200  feet  is  10.  We  have 

then  g  =  —  .16,  g'  —  .08,  and  n  —  10.     Hence  a  =  —  L  -  -~  —  = 

—  24 

'      =  —  .006.     The  first  grade  is,  therefore,  g  —  a  —  —  .16  -4- 

.006  =  —  .154,  and  as  each  subsequent  grade  increases  .012  (§  150), 
the  whole  may  be  written  down  without  farther  trouble,  thus  :  — 

-  .154,  -  JL42,  -  .130,  -  .118,  -  .106,'  -  .094,  -  .082,  -  .070, 

-  .058,  -  .046,  -  .034,  -  .022,  -  .010,   +  .002,    +  .014,   +  .026, 
+  .038,  +  .050,  +  .062,  +  .074. 

ARTICLE  IV.  —  ELEVATION  OF  THE  OUTER  RAIL  ON  CURVES. 

152.  Problem,  Given  the  radius  of  a  curve  R,  the  gauge  of 
the  track  g,  and  the  velocity  of  a  car  per  second  v,  to  determine  the 
proper  elevation  e  of  the  outer  rail  of  the  curve. 

Solution.  A  car  of  mass  M  moving  on  a  curve  of  radius  J?, 
with  a  velocity  per  second  =  v,  has,  by  Mechanics,  a  centrifugal 

M  v* 

force  =  —77-  .     To  counteract  this  force,  the  outer  rail  on  a  curve 
H 

is  raised  above  the  level  of  the  inner  rail,  so  that  the  car  may  rest 
on  an  inclined  plane.  This  elevation  must  be  such,  that  the  ac- 
tion of  gravity  in  forcing  the  car  down  the  inclined  plane  shall  be 
just  equal  to  the  centrifugal  force,  which  impels  it  in  the  opposite 
direction.  Now  the  action  of  gravity  on  a  body  resting  on  an  in- 
clined plane  is  equal  to  32.2  M  multiplied  by  the  ratio  of  the  height 
to  the  length  of  the  plane.  But  the  height  of  the  plane  is  the  ele- 
vation e,  and  its  length  the  gauge  of  the  track  g.  This  action  of 
gravity,  which  is  to  counteract  the  centrifugal  force,  is,  therefore, 

32  2  Me 

—  .     Putting  this  equal  to  the  centrifugal  force,  we  have 

32.2  Me      Mv* 

-  =  —  =—  .    Hence 

g  R 

eog-  r-    gv 

-32^ZT 

If  we  substitute  for  R  its  value  (§  10)  R  —  —  —  j.  ,  we  have  e  = 


~  -00062112  g  v*  sin.  D.    If  the  velocity  is  given  in  miles 


ELEVATION  OF  THE  OUTER  BAIL  ON  CURVES.      137 

V  x  5280 
per  hour,  represent  this  velocity  by  V,  and  we  have  v  —  -^— 

bl)  x  bO 

Substituting  this  value  of  v,  we  find  e  =  .0013361  g  F2  sin.  D. 
When  g  =  4.7,  this  becomes  e  —  .00627966  F2  sin.  D.  By  this  for- 
mula Table  VII.  is  calculated.  In  determining  the  proper  eleva- 
tion in  any  given  case,  the  usual  practice  is  to  adopt  the  highest 
customary  speed  of  passenger  trains  as  the  value  of  V. 

153.  Still  the  outer  rail  of  a  curve,  though  elevated  according 
to  the  preceding  formula,  is  generally  found  to  be  much  more 
worn  than  the  inner  rail.  On  this  account  some  are  led  to  distrust 
the  formula,  and  to  give  an  increased  elevation  to  the  raiL  So 
far,  however,  as  the  centrifugal  force  is  concerned,  the  formula  is 
undoubtedly  correct,  and  the  evil  in  question  must  arise  from 
other  causes, — causes  which  are  not  counteracted  by  an  additional 
elevation  of  the  outer  rail.  The  principal  of  these  causes  is  prob- 
ably improper  "  coning  "  of  the  wheels.  Two  wheels,  immovable 
on  an  axle,  and  of  the  same  radius,  must,  if  no  slip  is  allowed, 
pass  over  equal  spaces  in  a  given  number  of  revolutions.  Now  as 
the  outer  rail  of  a  curve  is  longer  than  the  inner  rail,  the  outer 
wheel  of  such  a  pair  must  on  a  curve  fall  behind  the  inner  wheel. 
The  first  effect  of  this  is  to  bring  the  flange  of  the  outer  wheel 
against  the  rail,  and  to  keep  it  there.  The  second  is  a  strain  on 
the  axle  consequent  upon  a  slip  of  the  wheels  equal  in  amount  to 
the  difference  in  length  of  the  two  rails  of  the  curve.  To  remedy 
this,  coning  of  the  wheels  was  introduced,  by  means  of  which  the 
radius  of  the  outer  wheel  is  in  effect  increased,  the  nearer  its 
flange  approaches  the  rail,  and  this  wheel  is  thus  enabled  to  trav- 
erse a  greater  distance  than  the  inner  wheel. 

To  find  the  amount  of  coning  for  a  play  of  the  wheels  of  one 
inch,  let  r  and  r'  represent  the  proper  radii  of  the  inner  and  outer 
wheels  respectively,  when  the  flange  of  the  outer  wheel  touches 
the  rail.  Then  r'  —  r  will  be  the  coning  for  one  inch  in  breadth 
of  the  tire.  To  enable  the  wheels  to  keep  pace  with  each  other  in 
traversing  a  curve,  their  radii  must  be  proportional  to  the  lengths 
of  the  two  rails  of  the  curve,  or,  which  is  the  same  thing,  propor- 
tional to  the  radii  of  these  rails.  If  R  be  taken  as  the  radius  of 
the  inner  rail,  the  radius  of  the  outer  rail  will  be  R  +  g,  and  we 
shall  have  r  :  r'  —  R  :  R  +  g.  Therefore,  rR  +  rg  =  r'  R,  or 


138  LEVELLING. 

As  an  example,  let  R  =  600,  r  =  1.4,  and  g  =  4.7.    Then  we 

14x47 
have  r'  —  r  =    '  =  .011  ft.    For  a  tire  3.5  in.  wide,  the  con- 

bUU 

ing  would  be  3.5  x  .011  =  .0385  ft.,  or  nearly  half  an  inch. 

Two  distinct  things,  therefore,  claim  attention  in  regard  to  the 
motion  of  cars  on  a  curve.  The  first  is  the  centrifugal  force, 
which  is  generated  in  all  cases,  when  a  body  is  constrained  to 
move  in  a  curvilinear  path,  and  which  may  be  effectually  counter- 
acted for  any  given  velocity  by  elevating  the  outer  rail.  The  sec- 
ond is  the  unequal  length  of  the  two  rails  of  a  curve,  in  conse- 
quence of  which  two  wheels  fixed  on  an  [axle  cannot  traverse  a 
curve  properly,  unless  some  provision  is  made  for  increasing  the 
diameter  of  the  outer  wheel.  Coning  of  the  wheels  was  devised 
for  this  purpose ;  but  as  the  coning,  when  at  all  considerable,  was 
found  to  produce  an  irregular  sidewise  motion  of  the  train,  the 
tendency  has  been  to  diminish  the  coning.  The  standard  wheel- 
tread  adopted  by  the  Master  Car  Builders'  Association  has  a  con- 
ing of  but  iV  of  an  inch  in  2|  inches  of  the  tread  next  to  the 
flange. 

ARTICLE  V.— EASING  GRADES  ON  CURVES. 


154.  When  a  curve  occurs  on  a  steep  grade  it  is  desirable  to 
ease  the  grade  on  the  curve,  so  as  to  make  the  joint  resistance  of 
the  grade  and  curve  equal  to  that  of  the  grade  alone  on  straight 
lines.  The  resistance  on  a  grade  is  proportional  to  the  rise  of  the 
grade  per  station  and  the  resistance  due  to  a  curve  can  be  repre- 
sented as  equivalent  to  that  of  a  grade  having  a  certain  rise  per 
station.  The  rise  per  station  of  the  eased  grade  will  be  simply 
the  original  rise  diminished  by  the  rise  that  represents  the  curve 
resistance.  The  resistance  caused  by  curves  varies  greatly  with 
the  state  of  the  track  and  the  kind  of  rolling  stock,  and  is  vari- 
ously estimated  as  equivalent  on  a  1°  curve  to  the  resistance  of  a 
grade  of  .025  to  .06  of  a  foot  per  station.  For  a  curve  of  any 
other  degree  the  resistance  increases  with  the  degree ;  so  that  a 
6°  curve,  for  example,  has  six  times  the  resistance  of  a  1°  curve. 
As  an  example  let  a  rise  of  .04  per  station  be  taken  as  the  resist- 
ance on  a  1°  curve  and  suppose  a  6°  curve  to  occur  on  a  grade  of 
1.6  per  station.  Then  the  reduced  grade  will  be  1.6  —  .24  —  1.36 
per  station. 


EXPANSION   OF  BAILS.  139 

ARTICLE  VI.— EXPANSION  OF  RAILS. 

155.  The  rails  of  a  track  exposed  to  a  summer  sun  may  rise  to  a 
temperature  of  130°  Fahrenheit.  When,  therefore,  a  track  is  laid 
at  a  much  lower  temperature,  as  is  usual,  provision  for  the  expan- 
sion of  the  rails  must  be  made  by  leaving  a  proper  space  between 
successive  rails.  The  expansion  of  a  bar  of  iron  or  steel  may  be 
taken  as  .000  007  of  its  length  for  every  degree  of  rise  in  tempera- 
ture. The  space  to  be  left  between  the  rails  will  vary  with  the 
length  of  the  rails  and  with  the  number  of  degrees  below  130° 
of  the  temperature  when  the  track  is  laid.  Suppose  30-feet  rails 
are  laid  at  a  temperature  of  50°.  Then  the  number  of  degrees  of 
possible  rise  of  temperature  is  130°  —  50°  =  80°,  and  the  space  to 
be  left  between  the  rails  is  .000  007  x  80  x  30  =  .0168  of  a  foot. 
In  general,  let  s  be*  the  space  to  be  left  between  the  rails,  n  the 
number  of  degrees  that  the  temperature  is  below  130°,  and  I  the 
length  of  the  rails  in  feet,  and  we  have 

s  =  .000  007  n  I. 

A  convenient  rule  for  30-feet  rails  may  be  obtained  by  putting 
in  the  formula  I  =  30  and  n  =  5,  whence,  nearly  enough,  8  =  .001. 
That  is,  the  space  to  be  left  is  one-thousandth  of  a  foot  for  every 
five  degrees  that  the  temperature  is  below  130°. 


140  EARTH-WORK. 

CHAPTER  V. 

EARTH-WORK. 
ARTICLE  I. — PRISMOIDAL  FORMULA. 

156.  EARTH-WORK  includes  the  regular  excavation  and  embank- 
ment on  the  line  of  a  road,  borrow-pits,  or  such  additional  excava- 
tions as  are  made  necessary  when  the  embankment  exceeds  the 
regular  excavation,  and,  in  general,  any-  transfers  of  earth  that 
require  calculation.     We  begin  with  the  prismoidal  formula,  as 
this  formula  is  frequently  used  in  calculating  cubical  contents 
both  of  earth  and  masonry. 

A  prismoid  is  a  solid  having  two  parallel  faces,  and  composed 
of  prisms,  wedges,  and  pyramids,  whose  common  altitude  is  the 
perpendicular  distance  between  the  parallel  faces. 

157.  Problem.     Given  the  areas  of  the  parallel  faces  B  and 
B ' ,  the  middle  area  M,  and  the  altitude  a  of  a  prismoid,  to  find 
its  solidity  S. 

Solution.  The  middle  area  of  a  prismoid  is  the  area  of  a  sec- 
tion midway  between  the  parallel  faces  and  parallel  to  them,  and 
the  altitude  is  the  perpendicular  distance  between  the  parallel 
faces.  If  now  b  represents  the  base  of  any  prism  of  altitude  a,  its 
solidity  is  a  b.  Ifb  represents  the  base  of  a  regular  wedge  or  half- 
parallelopipedon  of  altitude  a,  its  solidity  is  $  a  b.  If  b  represents 
the  base  of  a  pyramid  of  altitude  «,  its  solidity  is  £  a  b.  The  so- 
lidity of  these  three  bodies  admits  of  a  common  expression,  which 
may  be  found  thus :  Let  m  represent  the  middle  area  of  either  of 
these  bodies,  that  is,  the  area  of  a  section  parallel  to  the  base  and 
midway  between  the  base  and  top.  In  the  prism,  m  =  b,  in  the 
regular  wedge,  m  =  ^b9  and  in  the  pyramid,  m  =  $b.  Moreover, 
the  upper  base  of  the  prism  =  £,  and  the  upper  base  of  the  wedge 
or  pyramid  =  0.  Then  the  expressions  a  b,  -J-  a  b,  and  $  a  b  may  be 
thus  transformed.  Solidity  of 

prism       =     ab  =      x6b=(b  +  b  +  4b}  =     (b+b  +  4  m), 


BORROW-PITS.  141 


wedge      = 

pyramid  =  i  ab  =  |  x  26  =  |  (0  +  b  +  b)      =  |  (0  +  6  +  4  m). 

Hence,  the  solidity  of  either  of  these  bodies  is  found  by  adding 
together  the  area  of  the  upper  base,  the  area  of  the  lower  base, 
and  four  times  'the  middle  area,  and  multiplying  the  sum  by  one 
sixth  of  the  altitude.  Irregular  wedges,  or  those  not  half-paral- 
lelopipedons,  may  be  measured  by  the  same  rule,  since  they  are 
the  sum  or  difference  of  a  regular  wedge  and  a  pyramid  of  com- 
mon altitude,  and  as  the  rule  applies  to  both  these  bodies,  it  ap- 
plies to  their  sum  or  difference. 

Now  a  prismoid,  being  made  up  of  prisms,  wedges,  and  pyra- 
mids of  common  altitude  with  itself,  will  have  for  its  solidity  the 
sum  of  the  solidities  of  the  combined  solids.  But  the  sum  of  the 
areas  of  the  upper  and  lower  bases  of  the  combined  solids  is  equal 
to  B  +  B\  the  sum  of  the  areas  of  the  parallel  faces  of  the  pris- 
moid; and  the  sum  of  the  middle  areas  of  the  combined  solids  is 
equal  to  M,  the  middle  area  of  the  prismoid.  Therefore 


ARTICLE  II.  —  BORROW-PITS. 

158.  FOR  the  measurement  of  small  excavations,  such  as  borrow- 
pits,  &c.,  the  usual  method  of  preparing  the  ground  is  to  divide 
the  surface  into  parallelograms  *  or  triangles,  small  enough  to  be 
considered  planes,  laid  off  from  a  base  line,  that  will  remain  un- 
touched by  the  excavation.  A  convenient  bench-mark  is  then  se- 
lected, and  levels  taken  at  all  the  angles  of  the  subdivisions.  After 
the  excavation  is  made,  the  same  subdivisions  are  laid  off  from 
the  base  line  upon  the  bottom  of  the  excavation,  and  levels  re- 
ferred to  the  same  bench-mark  are  taken  at  all  the  angles. 

This  method  divides  the  excavation  into  a  series  of  vertical 
prisms,  generally  truncated  at  top  and  bottom.  The  vertical  edges 
of  these  prisms  are  known,  since  they  are  the  differences  of  the 

*  If  the  ground  is  divided  into  rectangles,  as  is  generally  done,  and  one 
side  be  made  27  feet,  or  some  multiple  of  27  feet,  the  contents  may  be  ob- 
tained at  once  in  cubic  yards,  by  merely  omitting  the  factor  27  in  the  calcu- 
lation. 


142 


EARTH -WORK. 


levels  at  the  top  and  bottom  of  the  excavation.  The  horizontal 
section  of  the  prisms  is  also  known,  because  the  parallelograms 
or  triangles,  into  which  the  surface  is  divided,  are  always  meas- 
ured horizontally. 

159.  Problem.  Given  the  edges  )i,  hi ,  and  h* ,  to  find  the 
solidity  S  of  a  vertical  prism,  whether  truncated  or  not,  whose 
horizontal  section  is  a  triangle  of  given  area  A. 


Fig.  68. 


Solution.  When  the  prism  is  not  truncated,  we  have  h  =  hi=. 
hi .  The  ordinary  rule  for  the  solidity  of  a  prism  gives,  therefore, 
S=  Ah  =  A  x  $(h  +  hi  +  h?).  When  the  prism  is  truncated,  let 
ABCFGH  (fig.  68)  represent  such  a  prism,  truncated  at  the 
top.  Through  the  lowest  point  A  of  the  upper  face  draw  a  hori- 
zontal plane  AD  E  cutting  off  a  pyramid,  of  which  the  base  is 
the  trapezoid  B  D  E  C,  and  the  altitude  a  perpendicular  let  fall 
from  A  on  D  E.  Represent  this  perpendicular  by^>,  and  we  have 
(Tab.  X.  52)  the  solidity  of  the  pyramid  =  lp  x  BDEC  —  Ipx 
DE  x  i(BD  +  CE)  =  $p  xDEx%(BD  +  C  E}  =  A  x  £ 
(BD  +  CE\  since  $p  xDE  =  ADE=A.  Eut^(BD  +  C  E) 
is  the  mean  height  of  the  vertical  edges  of  the  truncated  portion, 
the  height  at  A  being  0.  Hence  the  formula  already  found  for  a 
prism  not  truncated,  will  apply  to  the  portion  above  the  plane 
A  D  E,  as  well  as  to  that  below.  The  same  reasoning  would  ap- 


BORROW-PITS. 


143 


ply,  if  the  lower  end  also  were  truncated.    Hence,  for  the  solidity 
of  the  whole  prism,  whether  truncated  or  not,  we  have 

S=A  *k(h  +  hi  +  hi). 


160.  Problem.  Given  the  edges  h,  hi ,  h* ,  and  hs ,  to  find 
the  solidity  S  of  a  vertical  prism ,  whether  truncated  or  not,  whose 
horizontal  section  is  a  parallelogram  of  given  area  A. 

Solution.  Let  B  H  (fig.  69)  represent  such  a  prism,  whether 
truncated  or  not,  and  let  the  plane  B  F HD  divide  it  into  two 


Fig.  69, 


triangular  prisms  AFH  and  C FH.  The  horizontal  section  of 
each  of  these  prisms  will  be  •£ A,  and  if  h,  hi,  h*,  and  A$  repre- 
sent the  edges  to  which  they  are  attached  in  the  figure,  we  have 
for  their  solidity  (§  159)  AFH=$Ax$(h  +  hl  +  h»)9  and 
CFH=%A  x  $(hi  +  h?  +  h9).  Therefore,  the  whole  prism  will 
have  for  its  solidity  S  =  iAx^(h  +  2hl  +  hy  +  2AS).  Let  the 
whole  prism  be  again  divided  by  the  plane  A  E  G  C  into  two  tri- 
angular prisms  BEG  and  D  E  G.  Then  we  have  for  these  prisms, 
BEG—^A  x  $(h  +  Ai  +  7*2),  and  DEG  =  \A  x  $(h  +  h*  + 
A3),  and  for  the  whole  prism,  S  =  $A  x  ^(2h  +  h^  +2^2  +  h3). 
Adding  the  two  expressions  found  for  S,  we  have  2  S  =  $  A 
(h  +  hi  +  hi  +  h3),  or 

S  =  A  x  J  (h  f  hi  +  /la  +  hs). 


144 


EARTH-WOJRK. 


It  will  be  seen  by  the  figure,  that  i  (h  +  /i,2)  =  KL  =  $(hi  +  //3), 
or  Ji  +  h2  =  hi  -+-  7*3.  The  expression  for  S  might,  therefore,  be 
reduced  to  S  =  A  x  |  (h  +  /i2),  or  S  =  J.  x  |(/h  +  ^3).  But  as 
the  ground  surfaces  ABCD  and  E F Gr H  are  seldom  perfect 
planes,  it  is  considered  better  to  use  the  mean  of  the  four  heights, 
instead  of  the  mean  of  two  diagonally  opposite. 

161.  Corollary.  When  all  the  prisms  of  an  excavation  have 
the  same  horizontal  section  A,  the  calculation  of  any  number  of 
them  may  be  performed  by  one  operation.  Let  figure  70  be  a  plan 


a* 


b+ 


fa 


a. 


Fig.  70. 

of  such  an  excavation,  the  heights  at  the  angles  being  denoted  by 
a,  »i ,  a2 ,  b,  bi ,  &c.  Then  the  solidity  of  the  whole  will  be  equal 
to  I A  multiplied  by  the  sum  of  the  heights  of  the  several  prisms 
(§160).  Into  this  sum  the  corner  heights  a,  a*,  b,  b5,  c6,  d,  and 
d4  will  enter  but  once,  each  being  found  in  but  one  prism ;  the 
heights  «i ,  &4,  c,  di,  d^,  and  ds  will  enter  twice,  each  being  com- 
mon to  two  prisms ;  the  heights  bi ,  b3 ,  and  c4  will  enter  three 
times,  each  being  common  to  three  prisms ;  and  the  heights  &2 ,  Ci , 
C2 ,  and  c3  will  enter  four  times,  each  being  common  to  four  prisms. 
If,  therefore,  the  sum  of  the  first  set  of  heights  is  represented  by 
Si ,  the  sum  of  the  second  by  s2 ,  of  the  third  by  sa,  and  of  the 
fourth  by  s4 ,  we  shall  have  for  the  solidity  of  all  the  prisms 

S  =  i  A  («i  +  2  52  +  3  s3  +  4  «4). 


CENTRE   HEIGHTS   ALONE   GIVEN. 


145 


ARTICLE  III.—  EXCAVATION  AND  EMBANKMENT. 

162.  As  embankments  have  the  same  general  shape  as  excava- 
tions, it  will  be  necessary  to  consider  excavations  only.  The  sim- 
plest case  is  when  the  ground  is  considered  level  on  each  side  of 
the  centre  line.  Figure  71  represents  the  mass  of  earth  between 
two  stations  in-an  excavation  of  this  kind.  The  trapezoid  O  B  F  II 
is  a  section  of  the  mass  at  the  first  station,  and  Gl  Si  FI  Hi  a  sec- 
tion at  the  second  station  ;  A  E  is  the  centre  height  at  the  first 
station,  and  AI  EI  the  centre  height  at  the  second  station  ; 
H  Hi  Fi  F  is  the  road-bed,  G  Gi  BI  B  the  surface  of  the  ground, 
and  G  Gi  HI  H  and  B  BI  FI  F  the  planes  forming  the  side  slopes. 
This  solid  is  a  prismoid,  and  might  be  calculated  by  the  prismoid- 
al  formula  (§  157).  The  following  method  gives  the  same  result. 


A.  Centre  Heights  alone  given. 

163.  Problem.  Given  the  centre  heights  c  and  Ci  ,  the  width 
of  the  road-bed  &,  the  slope  of  the  sides  s,  and  the  length  of  the 
section  Z,  to  find  the  solidity  S  of  the  excavation. 

Solution.  Let  c  be  the  centre  height  at  A  (fig.  71)  and  Ci  the 
height  at  AI  .  The  slope  s  is  the  ratio  of  the  base  of  the  slope  to 


its  perpendicular  height  (§  144).  We  have  then  the  distance  out 
AB  —  \~b  +  sc,  and  the  distance  out  AI  BI  ==  |  b  +  s  Ci  (§  144). 
Divide  the  whole  mass  into  two  equal  parts  by  a  vertical  plane 
A  AI  EI  E  drawn  through  the  centre  line,  and  let  us  find  first  the 
11 


146  EARTH-  WORK. 

solidity  of  the  right-hand  half.  Through  B  draw  the  planes 
JBEEi,  BAiEi,  and  BEi$\,  dividing  the  half  -sect  ion  into 
three  quadrangular  pyramids,  having  for  their  common  vertex 
the  point  B,  and  for  their  bases  the  planes  A  A1E1EJ  EE1F1  F, 
and  AiBiFiEi.  For  the  areas  of  these  bases  we  have 

Area  of  A  A1  E^  E  =  i  E  E^  x  (A  E  +  Al  EJ    =  %l(c  +  d), 
•     «     "    E  EI  F^F  —EF  x  EEi  =  iH 

"     "   Al  JB,  Fl  Ei  =  iAl  El  x  (Ei  Fl  +  A:  B^}  =  |  (b  c,  +  s  d2), 

and  for  the  perpendiculars  from  the  vertex  B  on  these  bases,  pro- 
duced when  necessary, 

Perpendicular  on  A  AI  EI  E    = 


Then  (Tab.  X.  52)  the  solidities  of  the  three  pyramids  are 
sc)  x  ^?(c  +  c1)  =  i?(i&c  + 

S  C2  +  S  C  Ci 


B-A1B1F1E1  =  $1  x  i(6d  +  5C!2)  =     1  (b  d  +  s  d 

Their  sum,  or  the  solidity  of  the  half  -section,  is 

iS=-bl[$b(c  +  c,)  +  s(c2  +  d2  +  cci)]. 
Therefore  the  solidity  of  the  whole  section  is 


When  the  slope  is  1|  to  I,  s  =  f,  and  the  factor  f  s  =  1  may  be 
dropped. 

164.  Problem.  To  find  the  solidity  S  of  any  number  n  of 
successive  sections  of  equal  length. 

Solution.  Let  c,  Ci  ,  ca  ,  cs  ,  &c.,  denote  the  centre  heights  at  the 
successive  stations.  Then  we  have  (§  163) 

Solidity  of  first  section      —  £  Z  [6  (c   +  d)  +  f  5  (c2   +  d3  +  c  d)], 
"        "  second  section  =  %  I  [b  (d  +  ca)  +  f  s  (d2  +  ca2  +  d  e*)), 
"  third  section    =  i  I  [  b  (c,  +  c.)  +  f  s  (c22  +  c32  +  c2  c3)], 
&c.  &c. 

For  the  solidity  of  any  number  n  of  sections,  we  should  have  1  1 
multiplied  by  the  sum  of  the  quantities  in  n  parentheses  formed 


CENTRE   HEIGHTS   ALONE    GIVEN. 


147 


as  those  just  given.  The  last  centre  height,  according  to  the  nota- 
tion adopted,  will  be  represented  by  cn  ,  and  the  next  to  the  last 
by  cn—  i.  Collecting  the  terms  multiplied  by  b  into  one  line,  the 
squares  multiplied  by  f  s  into  a  second  line,  and  the  remaining 
terms  into  a  third  line,  we  have  for  the  solidity  of  n  sections 


(c  +  2  d  +  2  ca  +  2  c3  ..... 


S  = 


When  s  =  f  ,  the  factor  f  s  =  1  may  be  dropped. 


2  cn_i  +  cn) 
2  C*—!  +  c2n) 


Example.  Given  I  =  100,  b  =  28,  s  =  f ,  and  the  stations  and 
centre  heights  as  set  down  in  the  first  and  second  columns  of  the 
annexed  table.  The  calculation  is  thus  performed.  Square  the 
heights,  and  set  the  squares  in  the  third  column.  Form  the  suc- 
cessive products  c  Ci,  CxCa,  &c.,  and  place  them  in  the  fourth  col- 
umn. Add  up  the  last  three  columns.  To  the  sum  of  the  second 
column  add  the  sum  itself,  minus  the  first  and  the  last  height, 
and  to  the  sum  of  the  third  column  add  the  sum  itself,  minus  the 
first  and  the  last  square.  Then  86  is  the  multiplier  of  b  in  the 
first  line  of  the  formula,  592  is  the  second  line,  since  f  s  is  here  1, 
and  274  is  the  third  line.  The  product  of  86  by  b  =  28  is  2408, 
and  the  sum  of  274,  592,  and  2408  is  3274.  This  multiplied  by 
i  I  —  50  gives  for  the  solidity  163,700  cubic  feet. 


Station. 

c. 

c3. 

ccl. 

0 

2 

4 

1 

4 

16 

8 

2 

7 

49 

28 

3 

6 

36 

42 

4 

10 

100 

60 

5 

7 

49 

70 

6 

6 

36 

42 

7 

4 

16 

24 

~46      306  '   274  ' 

40     286     592 

28 


2408 


592         2408 
2)3274 
163700. 


148  EARTH-WORK. 

B.  Centre  and  Side  Heights  given. 

165.  When  greater  accuracy  is  required  than  can  be  attained  by 
the  preceding  method,  the  side  heights  and  the  distances  out 
(§  144)  are  introduced.  Let  figure  72  represent  the  right-hand 
side  of  an  excavation  between  two  stations.  AAiBiB  is  the 
ground  surface ;  A  E  =  c  and  AI  EI  =  GI  are  the  centre  heights ; 
B  G  =  h  and  Bl  G^  —  h^  the  side  heights ;  and  d  and  dl ,  the  dis- 
tances out,  or  the  horizontal  distances  of  B  and  BI  from  the  centre 
line.  The  whole  ground  surface  may  sometimes  be  taken  as  a 
plane,  and  sometimes  the  part  on  each  side  of  the  centre  line  may 
be  so  taken ;  *  but  neither  of  these  suppositions  is  sufficiently  ac- 
curate to  serve  as  the  basis  of  a  general  method.  In  most  cases, 
however,  we  may  consider  the  surface  on  each  side  of  the  centre 
line  to  be  divided  into  two  triangular  planes  by  a  diagonal  passing 
from  one  of  the  centre  heights  to  one  of  the  side  heights.  A  ridge 
or  depression  will,  in  general,  determine  which  diagonal  ought  to 
be  taken  as  the  dividing  line,  and  this  diagonal  must  be  noted  in 
the  field.  Thus,  in  the  figure  a  ridge  is  supposed  to  run  from  B 
to  A i ,  from  which  the  ground  slopes  downward  on  each  side  to  A 
and  BI  .  Instead  of  this,  a  depression  might  run  from  A  to  BI  , 
and  the  ground  rise  each  way  to  AI  and  B.  If  the  ridge  or  de- 
pression is  very  marked,  and  does  not  cross  the  centre  or  side  lines 
at  the  regular  stations,  intermediate  stations  must  be  introduced 
to  make  the  triangular  planes  conform  better  to  the  nature  of  the 
ground.  If  the  surface  happens  to  be  a  plane,  or  nearly  so,  the 
diagonal  may  be  taken  in  either  direction.  It  will  be  seen,  there- 
fore, that  the  following  method  is  applicable  to  all  ordinary 
ground.  When,  however,  the  ground  is  very  irregular,  the  method 
of  §  171  is  to  be  used. 

166.  Problem.  Given  the  centre  heights  c  and  Ci ,  the  side 
heights  on  the  right  h  and  h\ ,  on  the  left  h1  and  h'i ,  the  distances 
out  on  the  right  d  and  di ,  on  the  left  d'  and  d\ ,  the  width  of  the 


*  It  is  easy  in  any  given  case  to  ascertain  whether  a  surface  like  A  Al  Bl  B 
is  a  plane  ;  for  if  it  is  a  plane,  the  descent  from  A  to  B  will  be  to  the  de- 
scent from  Al  to  1?,,  as  the  distance  out  at  the  first  station  is  to  the  distance 
out  at  the  second  station  ;  that  is,  c  -  h  :  ct  —  hl  =  d  :  dl.  If  we  had  c  =  9, 
h  =  6,  Ci  =  12.  hi  =  8,  d  =  24,  and  c?j  =  27,  the  formula  would  give  3:4  = 
24  :  27,  which  shows  that  the  surface  is  not  a  plane. 


CENTRE    AND    SIDE    HEIGHTS    GIVEN. 


149 


road-bed  b,  the  length  of  the  section  Z,  and  the  direction  of  the 
diagonals,  to  find  the  solidity  S  of  the  excavation. 

Solution.  Let  figure  72  represent  the  right-hand  side  of  the 
excavation,  and  let  us  suppose  first,  that  the  diagonal  runs,  as 
shown  in  the  figure,  from  B  to  AI.  Through  B  draw  the  planes 
BEE^  BAiEt,  and  BE1F1,  dividing  the  half-section  into 
three  quadrangular  pyramids,  having  for  their  common  vertex 
the  point  B,  and  for  their  bases  the  planes  A  AI  EI  E,  E  E\  FI  Fy 
and  AiBiFiJZi.  For  the  areas  of  these  bases  we  have 

Areaof  AA1E1E    =  %EEi  x  (AE  +  A^Ei)      =iZ(c  +  Ci), 
^F   =EF  x 


and  for  the  perpendiculars  from  the  vertex  B  on  these  bases,  pro- 
duced when  necessary, 

Perpendicular  on  A  Al  El  E    =  E  6r  =  d, 

=h, 


A  . 


Fig.  72. 


V"~" :G 

Then  (Tab.  X.  52)  the  solidities  of  the  three  pyramids  are 


Their  sum,  or  the  solidity  of  the  half-section,  is 


(1) 


150  EARTH-  WORK. 

Next,  suppose  that  the  diagonal  runs  from  A  to  Bi  .  In  this 
case,  through  BI  draw  the  planes  B^EiE,  BiAE,  and  BtE  F 
(not  represented  in  the  figure),  dividing  the  half-section  again 
into  three  quadrangular  pyramids,  having  for  their  common  ver- 
tex the  point  B\,  and  for  their  bases  the  planes  A  AI  EI  E, 
E  EI  FL  jP,  and  A  B  F  E.  For  the  areas  of  these  bases  we  have 


Area  of  AAiEiE=$EEi  x  (A  E  +  A,  E,}  =  $l(c  + 
«     «  E  E\FiF  =  EF  x  EEl  =  iK 

"     "  ABFE    =  %AE  x  d  +  \EF  x  h  =  $dc  + 

and  for  the  perpendiculars  from  BI  on  these  bases,  produced  when 
necessary, 

Perpendicular  on  A  AI  Et  E  —  EI  G1  =  d1} 
"  E  ElFiF  =  Bi  Gi  =  hi, 
"  "  ABFE    =  E  Ei  =  I 

Then  (Tab.  X.  52)  the  solidities  of  the  three  pyramids  are 

Bi  —  A  A!  Ei  E  =  £  di  x  il  (c  +  d)       =  £  I  (di  c  +  dl  d), 

Bi  —  EE^Fi  F=%lil  x  i  ~bl  —lllh^ 

Bi-  ABFE    =$1    x  1  (d  c  +  1  b  h)  =  J  Z  (d  c  +  i  5  ft). 

Their  sum,  or  the  solidity  of  the  half  -section,  is 

+  di  d  +  di  c  +  b  hi  +  i  ~b  h).  (2) 


We  have  thus  found  the  solidity  of  the  half  -section  for  both  di- 
rections of  the  diagonal.  Let  us  now  compare  the  results  (1)  and 
(2),  and  express  them,  if  possible,  by  one  formula.  For  this  pur- 
pose let  (1)  be  put  under  the  form 

£  I  [d  c  +  di  d  +  d  d  +  |  &  (h  +  Ai  +  h)], 
and  (2)  under  the  form 

£  Z  [d  c  +  di  Ci  +  di  c  +  i  b  (h  +  ^i  +  7^)]. 

The  only  difference  in  these  two  expressions  is,  that  d  ^  and  the 
last  h  in  the  first,  become  dl  c  and  hi  in  the  second.  But  in  the 
first  case  c\  and  h  are  the  heights  at  the  extremities  of  the  diago- 
nal, and  d  is  the  distance  out  corresponding  to  h  ;  and  in  the  sec- 
ond case  c  and  hi  are  the  heights  at  the  extremities  of  the  diago- 
nal, and  di  is  the  distance  out  corresponding  to  hi.  Denote  the 
centre  height  touched  by  the  diagonal  by  C,  the  side  height  touched 
by  the  diagonal  by  H,  and  the  distance  out  corresponding  to  the 


CENTRE   AND   SIDE   HEIGHTS   GIVEN. 


151 


aide  height  H  by  D.  We  may  then  express  both  d  c±  and  di  c  by 
D  (7,  and  both  h  and  h^  by  H\  so  that  the  solidity  of  the  half- 
section  on  the  right  of  the  centre  line,  whichever  way  the  diago- 
nal runs,  may  be  expressed  by 


DC  + 


(3) 


To  obtain  the  contents  of  the  portion  on  the  left  of  the  centre 
line,  we  designate  the  quantities  on  the  left  by  the  same  letters 
used  for  corresponding  quantities  on  the  right,  merely  attaching 
a  0  to  them  to  distinguish  them.  Thus  the  side  heights  are  h' 
and  h\,  and  the  distances  out  d'  and  d\  ,  while  Z>,  (7,  and  H  be- 
come D',  C',  and  //'.  The  solidity  of  the  half  -section  on  the 
left  may  therefore  be  taken  directly  from  (3),  which  will  become 


D'C' 


(4) 


Finally,  by  uniting  (3)  and  (4),  we  obtain  the  following  formula 
for  the  solidity  of  the  whole  section  between  two  stations  : 


S= 


d')e 


D  C+D'  C1 


Example.  Given  I  —  100,  b  =  18,  and  the  remaining  data,  as 
arranged  in  the  first  six  columns  of  the  following  table.  The  first 
column  gives  the  stations;  the  fourth  gives  the  centre  heights, 
namely,  c  =  13.6  and  c\  —  8  ;  the  two  columns  on  the  left  of  the 
centre  heights  give  the  side  heights  and  distances  out  on  the  left 
of  the  centre  line  of  the  road,  and  the  two  columns  on  the  right 
of  the  centre  heights  give  the  side  heights  and  distances  out  on 
the  right.  The  direction  of  the  diagonals  is  marked  by  the 
oblique  lines  drawn  from  h'  =  8  to  Ci  =  8  and  from  c  =  13.6  to 
h,  =  12. 


Sta.  d 

'.   h'.     c 

h.   d. 

d  +  d'. 

(d  +  d')c.  D' 

C'.   D  C. 

0 
1 

21  8\  IS 
15  4  ^  8 

.6^   10  2 
.0  ^12  2 

4    45 
7    42 

612 
336    1 

168  367.2 

12 

12 

168 

20 

367.2 

54  x  ( 

1  = 

486 

6)1969.20 

32820. 

152  EARTH-  WORK. 

To  apply  the  formula,  the  distances  out  at  each  station  are 
added  together,  and  their  sum  placed  in  the  seventh  column  ; 
these  sums,  multiplied  by  the  respective  centre  heights,  are  placed 
in  the  eighth  column  ;  the  product  of  d'  =  21  (which  is  the  distance 
out  corresponding  to  the  side  height  touched  by  the  left-hand  diag- 
onal) by  d  =  8  (which  is  the  centre  height  touched  by  the  same 
diagonal)  is  placed  in  the  ninth  column,  and  the  similar  product 
of  d^—  27  by  c  =  13.6  is  placed  in  the  last  column.  The  terms  in 
the  formula  multiplied  by  £  b  are  all  the  side  heights,  and  in  ad- 
dition all  the  side  heights  touched  by  diagonals,  or  8  4-  4  +  10  + 
12  +  8  +  12  =  54.  Then  by  substitution  in  the  formula,  we  have 
S  =  $  x  100  (612  +  336  +  168  +  367.2  +  9  x  54)  =  32,820  cubic 
feet, 

By  applying  the  rule  given  in  the  note  to  §  165,  we  see  that 
the  surface  on  the  left  of  the  centre  line  in  the  preceding  ex- 
ample is  a  plane  ;  since  13.6  —  8  :  8  —  4  =  21  :  15.  The  diagonal 
on  that  side  might,  therefore,  be  taken  either  way,  and  the  same 
solidity  would  be  obtained.  This  may  be  easily  seen  by  revers- 
ing the  diagonal  in  this  example,  and  calculating  the  solidity 
anew.  The  only  parts  of  the  formula  affected  by  the  change 
are  D'  C"  and  \IH  '.  In  the  one  case  the  sum  of  these  terms  is 
21  x  8  +  9  x  8,  and  in  the  other  15  x  13.6  +  9x4,  both  of  which 
are  equal  to  240. 

167.  Problem.  To  find  the  solidity  S  of  any  number  n  of 
successive  sections  of  equal  length. 

Solution.  Let  c,  d  ,  c2  ,  c3  ,  &c.,  be  the  centre  heights  at  the  suc- 
cessive stations;  h,  7^,  Aa,  ha,&c.9  the  right-hand  side  heights;  h', 
h'i  ,  h'z  ,  h'a,  &c.,  the  left-hand  side  heights  ;  dtdltd9ldai  &c.,  the 
distances  out  on  the  right;  and  d',  d'L,  d'9,  d'3,&c.,  the  distances 
out  on  the  left.  Then  the  formula  for  the  solidity  of  one  section 
(g  166)  gives  for  the  solidities  of  the  successive  sections 

%l[(d  +  d')  c  +  (dl  +  d\)  d  +  D  C  +  D'  C'  +  i  b  (h  +  h,  +  H  + 
hf  +  h',  +  H')l 


i  I  [(d,  +  d'3)  c*  +  (d,  +  d'9)  c,  +  DiC*  +  D\  C\ 
//a  +  h'i  +  h'3  +  /J'2)], 

and  so  on,  for  any  number  of  sections.    For  the  solidity  of  any 


CENTRE   AND   SIDE   HEIGHTS   GIVEN. 


153 


number  n  of  sections,  we  should  have  £  I  multiplied  by  the  sum  of 
n  parentheses  formed  as  those  just  given.     Hence 


DC  +  DC1  +  DiCi-\-D'l( 

•ib   h  +  2 hj.  +  2 ^ + 


"i  +  //2  +  &c. 


Example.    Given  I  =  100,  b  =  28,  and  the  remaining  data  as 
given  in  the  first  six  columns  of  the  following  table : 


Sta. 

d'. 

fc'. 

c. 

h. 

d. 

d  +  d'. 

(d  +  d')  c. 

D'C'. 

DC. 

0 

17 

2\ 

2\ 

2 

17 

34 

68 

1 

18.5 

3 

>4\ 

^5 

21.5 

40 

160 

68 

43 

2 

20 

4^ 

^5\ 

"-6 

23 

43 

215 

80 

92 

3 

23 

(K 

/6\ 

^fl 

26 

49 

294 

115 

130 

4 

21.5 

5^ 

2* 

>7 

24.5 

46 

276 

129 

147 

5 

20 

4-^ 

^^ 

^4 

20 

40 

240 

120 

147 

6 

15.5 

!>- 

4^ 

3 

18.5 

34 

136 

93 

80 

25       35             1389   605  639 

22       30             1185 

22       37              605 

69      102              639 

102                     2394 

171  x  14  =  2394           6)6212 

103533  cubic  feet. 

The  data' in  this  table  are  arranged  precisely  as  in  the  example 
for  calculating  one  section  (§  166),  and  the  remaining  columns  are 
calculated  as  there  shown.  Then,  to  obtain  the  first  line  of  the 
formula,  add  all  the  numbers  in  the  column  headed  (d  +  d')  c, 
making  1389,  and  afterwards  all  the  numbers  except  the  first  and 
the  last,  making  1185.  The  next  line  of  the  formula  is  the  sum 
of  the  columns  D'C'  and  DC,  which  give  respectively  605  and 
639.  To  obtain  the  first  line  of  the  quantities  multiplied  by  £  &, 
add  all  the  numbers  in  column  ^,  making  35,  next  all  the  numbers 
except  the  first  and  the  last,  making  30,  and  lastly  all  the  numbers 
touched  by  diagonals  (doubling  any  one  touched  by  two  diago- 
nals), making  37.  The  second  line  of  the  quantities  multiplied  by 
^  &  is  obtained  in  the  same  way  from  the  column  marked  h'.  The 
sum  of  these  numbers  is  171,  and  this  multiplied  by  £  5  =  14  gives 


154  EARTH- WORK. 

2394.    We  have  now  for  the  first  line  of  the  formula  1389  +  1185, 
for  the  second  605  +  639,  and  for  the  remainder  2394.    By  adding 

100 
these  together,  and  multiplying  the  sum  by  £  I  =  ~—~ ,  we  get  the 

contents  of  the  six  sections  in  feet. 

168.  When  the  section  is  partly  in  excavation  and  partly  in 
embankment,  the  preceding  formulae  are  still  applicable ;  but  as 
this  application  introduces  minus  quantities  into  the  calculation, 
the  following  method,  similar  in  principle,  is  preferable. 

169.  Problem.     Given  the  widths  of  an  excavation  at  the 
road-bed  AF=w  and  AiFi=Wi  (fig.  73),  the  side  heights  h 
and  hi ,  the  length  of  the  section  Z,  and  the  direction  of  the  diago- 
nal, to  find  the  solidity  S  of  the  excavation,  when  the  section  is 
partly  in  excavation  and  partly  in  embankment. 


Solution.  Suppose,  first,  that  the  surface  is  divided  into  two 
triangles  by  the  diagonal  B  A  \  .  Through  B  draw  the  plane 
B  AI  Fi  ,  dividing  that  part  of  the  section  which  is  in  excavation 
into  two  pyramids  B  —  AAi  Fl  F  and  B  —  Al  B^  F^  ,  the  solidi- 
ties of  which  are 

B- 


The  whole  solidity  is,  therefore, 

S  =  i  I  (w  h  +  wl  7h  +  Wi  h). 

Next,  suppose  the  dividing  diagonal  to  run  from  A  to  BI  . 
Through  BI  draw  a  plane  BiAF  (not  represented  in  the  figure), 
dividing  the  excavation  again  into  two  pyramids,  of  which  the 
solidities  are 


CENTRE   AND    SIDE    HEIGHTS   GIVEN. 


155 


Bl  —  A  Ai  F!  F  — 
Bi-ABF        = 


x  i  I  (w  + 
x 


+  wl  hi), 


The  whole  solidity  is,  therefore, 

S  =  £  I  (w  h  +  Wi  Jii  +w  hi). 

The  only  difference  in  these  two  expressions  is,  that  w\  h  in  the 
first  becomes  whi  in  the  second.  But  in  the  first  case  the  diago- 
nal touches  Wi  and  A,  and  in  the  second  case  it  touches  w  and  ^i  . 
If,  then,  we  designate  the  width  touched  by  the  diagonal  by  TF, 
and  the  height  touched  by  the  diagonal  by  H,  we  may  express 
both  wih  and  wht  by  WH\  so  that  the  solidity  in  either  case 
may  be  expressed  by 

S  =  %l  (w  h  +  Wi  ^  +  WH). 


Corollary.  When  several  sections  of  equal  length  succeed 
one  another,  the  whole  may  be  calculated  together.  For  this  pur- 
pose, the  preceding  formula  gives  for  the  solidities  of  the  succes- 
sive sections 


1  (Wi  ^  +  W*  hi  +  TTi  Hi), 
J  I  (wi  h^  +  w9  h-3  +  Wi  Hi), 

and  so  on  for  any  number  of  sections.    Hence  for  the  solidity  of 
any  number  n  of  sections  we  should  have 


Wi  Hi  +  &c.) 

Example.    Given  I  =  100,  and  the  remaining  data  as  given  in 
the  first  three  columns  of  the  following  table  : 


Station. 

w. 

fc. 

wh. 

WH. 

0 

2 

/I 

2 

1 

8< 

6 

48 

8 

2 

10\ 

\7 

70 

56 

3 

13^ 

>s? 

91 

70 

4 

9 

\4 

36 

52 

247           186 

209 

186 

6)642 

10700. 


The  fourth  column  contains  the  products  of  the  several  widths 
by  the  corresponding  heights,  and  the  next  column  the  products 


156 


EARTH-WOKK. 


of  those  widths  and  heights  touched  by  diagonals,  The  sum  of 
the  products  in  the  fourth  column  is  247,  the  sum  of  all  but  the 
first  and  the  last  is  209,  and  the  sum  of  the  products  in  the  fifth 
column  is  186.  These  three  sums  are  added  together,  multiplied 
by  100,  and  divided  by  6,  according  to  the  formula.  This  gives  the 
solidity  of  the  four  sections  —  10700  cubic  feet. 

170.  When  the  excavation  does  not  begin  on  a  line  at  right  an- 
gles to  the  centre  line,  intermediate  stations  are  taken  where  the 
excavation  begins  on  each  side  of  the  road-bed,  and  the  section 
may  be  calculated  as  a  pyramid,  having  its  vertex  at  the  first  of 
these  points,  and  for  its  base  the  cross-section  at  the  second.    The 
preceding  method  gives  the  same  result,  since  w  and  h  in  this 
case  become  0,  and  reduce  the  formula  to  S  =  ^lwihi.     The 
same  remarks  apply  to  the  end  of  an  excavation. 

C.  Ground  very  Irregular. 

171.  Problem.     To  find  the  solidity  of  a  section,  when  the 
ground  is  very  irregular. 

Solution.  Let  A  HB F E  -  Al  CD  Bl  F^  El  (fig.  74)  represent 
one  side  of  a  section,  the  surface  of  which  is  too  irregular  to  be 
divided  into  two  planes.  Suppose,  for  instance,  that  the  ground 


Fig.  74. 


GROUND  VERY  IRREGULAR.  157 

changes  at  IT,  C,  and  D,  making  it  necessary  to  divide  the  surface 
into  five  triangles  running  from  station  to  station.*  Let  heights 
be  taken  at  H,  C,  and  D,  and  let  the  distances  out  of  these  points 
be  measured.  If  now  we  suppose  the  earth  to  be  excavated  verti- 
cally downward  through  the  side  line  B  BI  to  the  plane  of  the 
road-bed,  we  may  form  as  many  vertical  triangular  prisms  as 
there  are  triangles  on  the  surface.  This  will  be  made  evident  by 
drawing  vertical  planes  through  the  sides  A  C,  II C,  HD,  and 
HBi .  Then  the  solidity  of  the  half-section  will  be  equal  to  the 
sum  of  these  prisms,  minus  the  triangular  mass  B  F  G  —  B^Fi  G\. 

The  horizontal  section  of  the  prisms  may  be  found  from  the 
distances  out  and  the  length  of  the  section,  and  the  vertical  edges 
or  heights  are  all  known.  Hence  the  solidities  of  these  prisms 
may  be  calculated  by  §  159. 

To  find  the  solidity  of  the  portion  BFG  —  B1F1G1,  which  is 
to  be  deducted,  represent  the  slope  of  the  sides  by  s  (§  144),  the 
heights  at  B  and  BI  by  h  and  hi ,  and  the  length  of  the  section  by 
1.  Then  we  have  F  G  =  sh,  and  FI  Gi  =  s  h\ .  Moreover,  the 
area  of  BFG  =  $sh\  and  that  of  Bl  F1  Ol  =  ish^.  Now  as 
the  triangles  B  F  G  and  Bl  FI  G^  are  similar,  the  mass  required  is 
the  frustum  of  a  pyramid,  and  the  mean  area  is  y\  sh*  x  -£  s  h^  = 
Then  (Tab.  X.  53)  the  solidity  is  B  F  G  -  BI  Fl  G i  = 


Example.  Given  I  —  50,  b  =  18,  s  —  f ,  the  heights  at  A,  H,  and 
B  respectively  4,  7,  and  6,  the  distances  A  H  —  9  and  HB  =  9, 
the  heights  at  AI  ,  (7,  D,  and  BI  respectively  6,  7,  9,  and  8,  and  the 
distances  A  i  C  =  4,  CD  =  5,  and  DBt  =  12.  Then  the  horizon- 
tal section  of  the  first  prism  adjoining  the  centre  line  is  1 1  x  A  i  (?, 
since  the  distance  A\  C  is  measured  horizontally;  and  the  mean 
of  the  three  heights  is  £  (4  +  6  +  7)  =  i  x  17.  The  solidity  of 
this  prism  is  therefore  -J- 1  x  AI  C  x  £  x  17  =  \  I  x  4  x  17,  that  is, 
equal  to  £  I  multiplied  by  the  base  of  the  triangle  and  by  the  sum 
of  the  heights.  In  this  way  we  should  find  for  the  solidity  of  the 
five  prisms 

\l  (4  x  17  +  9  x  18  +  5  x  23  +  12  x  24  +  9  x  21)  =  \  I  x  822. 


*  It  will  often  be  necessary  to  introduce  intermediate  stations,  in  order  to 
make  the  subdivision  into  triangles  more  conveniently  and  accurately. 


158  EARTH- WORK. 

For  the  frustum  to  be  deducted,  we  have 

i  Z  x  f  (62  +  82  4-  6  x  8)  =  $1  x  222. 
Hence  the  solidity  of  the  half-section  is 

£  I  (822  -  222)  =  $  x  50  x  600  =  5000  cubic  feet. 

172.  Let  us  now  examine  the  usual  method  of  calculating  ex- 
cavation, when  the  cross-section  of  the  ground  is  not  level.  This 
method  consists,  first,  in  finding  the  area  of  a  cross-section  at  each 
end  of  the  mass;  secondly,  in  finding  the  height  of  a  section, 
level  at  the  top,  equivalent  in  area  to  each  of  these  end  sections ; 
thirdly,  in  finding  from  the  average  of  these  two  heights  the  mid- 
dle area  of  the  mass ;  and,  lastly,  in  applying  the  prisraoidal  for- 
mula to  find  the  contents.  The  heights  of  the  equivalent  sections 
level  at  the  top  may  be  found  approximately  by  Trautwine's  Dia- 
grams,* or  exactly  by  the  following  method.  Let  A  represent 
the  area  of  an  irregular  cross-section,  b  the  width  of  the  road-bed, 
and  s  the  slope  of  the  sides.  Let  x  be  the  required  height  of  an 
equivalent  section  level  at  the  top.  The  bottom  of  the  equivalent 
section  will  be  &,  the  top  b  +  2  s  x,  and  the  area  will  be  the  sum 
of  the  top  and  bottom  lines  multiplied  by  half  the  height  or 
£  x  (2  b  +  2  s  x}  =  s  x*  +  bx.  But  this  area  is  to  be  equal  to  A. 
Therefore,  sx*  +  bx  =  A,  and  from  this  equation  the  value  of  x 
may  be  found  in  any  given  case. 

According  to  this  method,  the  contents  of  the  section  already 
calculated  in  §  166  will  be  found  thus.  Calculating  the  end  areas, 
we  find  the  first  end  area  to  be  387  and  the  second  to  be  240. 
Then  as  s  is  here  f  and  b  =  18,  the  equations  for  finding  the 
heights  of  the  equivalent  end  sections  will  be  f  x*  +  18  x  =  387, 
and  f  x*  +  18  x  —  240.  Solving  these  equations,  we  have  for  the 
height  at  the  first  station  x  =  11.146,  and  at  the  second,  x  =  8. 
The  middle  area  will,  therefore,  have  the  height  -£(11.146  +  8)  = 
9.573,  and  from  this  height  the  middle  area  is  found  to  be  309.78. 
Then  by  the  prismoidal  formula  (§  157)  the  solidity  will  be  S  = 
fc  x  100  (387  +  240  +  4  x  309.78)  =  31102  cubic  feet. 

But  the  true  solidity  of  this  section  was  found  to  be  32820  cubic 
feet,  a  difference  of  1718  feet.  The  error,  of  course,  is  not  in  the 
prismoidal  formula,  but  in  assuming  that,  if  the  earth  were  levelled 

*  A  New  Method  of  Calculating  the  Cubic  Contents  of  Excavations  and 
Embankments  by  the  aid  of  Diagrams.  By  John  C.  Trautwine. 


CORRECTION  IN  EXCAVATION  ON  CURVES. 


159 


at  the  ends  to  the  height  of  the  equivalent  end  sections,  the  inter- 
vening earth  might  be  so  disposed  as  to  form  a  plane  between 
these  level  ends,  thus  reducing  the  mass  to  a  prismoid.  This  sup- 
position, however,  may  sometimes  be  very  far  from  correct,  as  has 
just  been  shown.  If  the  diagonal  on  the  right-hand  side  in  this 
example  were  reversed,  that  is,  if  the  dividing  line  were  formed 
by  a  depression,  the  true  solidity  found  by  §  166  would  be  29600 
feet ;  whereas  the  method  by  equivalent  sections  would  give  the 
same  contents  as  before,  or  1502  feet  too  much. 

D.  Correction  in  Excavation  on  Curves.  * 

173.  In  excavations  on  curves  the  vertical  planes  forming  the 
ends  of  a  section  are  not  parallel  to  each  other,  but  converge 
towards  the  centre  of  the  curve.  A  section  between  two  stations 
100  feet  apart  on  the  centre  line  will,  therefore,  measure  less  than 
100  feet  on  the  side  nearest  to  the  centre  of  the  curve,  and  more 
than  100  feet  on  the  side  farthest  from  that  centre.  Now  in 
calculating  the  contents  of  an  excavation,  it  is  assumed  that  the 
ends  of  a  section  are  parallel,  both  being  perpendicular  to  the 


B,    B 


Fig.  75. 


chord  of  the  curve.  Thus,  let  figure  75  represent  the  plan  of 
two  sections  of  an  excavation,  EF  G  being  the  centre  line,  A  L 
and  C M  the  extreme  side  lines,  and  0  the  centre  of  the  curve. 


160  EARTH- WORK. 

Then  the  calculation  of  the  first  section  would  include  all  be- 
tween the  lines  AI  Ci  and  BiDi',  while  the  true  section  lies 
between  A  C  and  B  D.  In  like  manner,  the  calculation  of  the 
second  section  would  include  all  between  H K  and  N P,  while 
the  true  section  lies  between  B  D  and  L  M.  It  is  evident,  there- 
fore, that  at  each  station  on  the  curve,  as  at  F,  the  calculation 
is  too  great  by  the  wedge-shaped  mass  represented  by  KFDi, 
and  too  small  by  the  mass  represented  by  B^  F  H.  These  masses 
balance  each  other,  when  the  distances  out  on  each  side  of  the 
centre  line  are  equal,  that  is,  when  the  cross-section  may  be 
represented  by  A  D  F  R  E  (fig.  76).  But  if  the  excavation  is 
on  the  side  of  a  hill,  so  that  the  distances  out  differ  very 


Fig.  76. 


much,  and  the  cross-section  is  of  the  shape  A  D  F  B  E,  the 
difference  of  the  wedge-shaped  masses  may  require  considera- 
tion. 

174.  Problem.  Given  the  centre  height  c,  the  greatest  side 
height  h,  the  least  side  height  h,  the  greatest  distance  out  d,  the 
least  distance  out  d',  and  the  width  of  the  road-bed  b,  to  find  the 
correction  in  excavation  C,  at  any  station  on  a  curve  of  radius  Tir 
or  deflection  angle  D. 

Solution.  The  correction,  from  what  has  been  said  above,  is  a 
triangular  prism  of  which  B  F  R  (fig.  76)  is  a  cross-section.  The 
height  of  this  prism  at  B  (fig.  75)  is  Bl  H,  the  height  at  R  is  R,  S, 
and  the  height  at  F  is  0.  BI  H  and  RI  S,  being  very  short,  are 
here  considered  straight  lines.  Now  we  have  the  cross-section 


—  h).  To  find  the  height  B^  H,  we  have  the 
angle  BFH=  BFBl  =  D,  and  therefore  B^  H—  2  HFsin.  D  = 
2  d  sin.  D.  In  like  manner,  7^  S  =  K  Dl  =  2  K  F  sin.  D  = 
2d'  sin.  D.  Then  since  the  height  at  F  is  0,  one  third  of  the  sum 
of  the  heights  of  the  prism  will  be  f  (d  +  d')  sin.  D,  and  the  cor- 
rection, or  the  solidity  of  the  prism,  will  be  (§  159) 


CORRECTION  IN  EXCAVATION  ON  CURVES.       161 

—  h)]  x  f  (d  +  d')  sin.  D. 


When  R  is  given,  and  not  D,  substitute  for  sin.  D  its  value  (§  9) 
sin.  D  —  -^  .    The  correction  then  becomes 


. 

This  correction  is  to  be  added,  when  the  highest  ground  is  on 
the  convex  side  of  the  curve,  and  subtracted,  when  the  highest 
ground  is  on  the  concave  side.  At  a  tangent  point,  it  is  evident, 
from  figure  75,  that  the  correction  will  be  just  half  of  that  given 
above. 

Example.  Given  e  =  28,  h  =  40,  h'  -  16,  d  =  74,  d  =  3S,b  = 
28,  and  R  =  1400,  to  find  G.  Here  the  area  of  the  cross-section 

OQ  OQ 

B  FR  =  Y  (74  -  38)  +  ~  (40  -  16)  =  672,  and  one  third  of  the 

.    100  (74  +  38)      8      „ 
sum  of  the  heights  of  the  prism  is  ^     =  -  .    Hence  C  — 

_  o  X  1.4UU  o 

672  x  |  =  1792  cubic  feet. 

o 

175.  When  the  section  is  partly  in  excavation  and  partly  in  em- 
bankment, the  cross-section  of  the  excavation  is  a  triangle  lying 
wholly  on  one  side  of  the  centre  line,  or  partly  on  one  side  and 
partly  on  the  other.     The  surface  of  the  ground,  instead  of  ex- 
tending from  B  to  D  (fig.  76),  will  extend  from  B  to  a  point  be- 
tween G  and  E,  or  to  a  point  between  A  and  G.    In  the  first  case, 
the  correction  will  be  a  triangular  prism  lying  between  the  lines 
B!  F  and  H  F  (fig.  75),  but  not  extending  below  the  point  F. 
In  the  second  case,  the  excavation  extends  below  F,  and  the  cor- 
rection, as  in  §  173,  is  the  difference  between  the  masses  above  and 
below  F.    This  difference  may  be  obtained  in  a  very  simple  man- 
ner, by  regarding  the  mass  on  both  sides  of  F  as  one  triangular 
prism  the  bases  of  which  intersect  on  the  line  G  F  (fig.  76),  in 
which  case  the  height  of  the  prism,  at  the  edge  below  F  must  be 
considered  to  be  minus,  since  the  direction  of  this  edge,  referred 
to  either  of  the  bases,  is  contrary  to  that  of  the  two  others.    The 
solidity  of  this  prism  will  then  be  the  difference  required. 

176.  Problem.     Given  the  width  of  the  excavation  at  the 
road-bed  w,  the  width  of  the  road-bed  b,  the  distance  out  d,  and 

12 


162  EARTH-WORK. 

the  side  height  h,  to  find  the  correction  in  excavation  (7,  at  any 
station  on  a  curve  of  radius  R  or  deflection  angle  D,  when  the 
section  is  partly  in  excavation  and  partly  in  embankment. 

Solution.  When  the  excavation  lies  wholly  on  one  side  of  the 
centre  line,  the  correction  is  a  triangular  prism  having  for  its 
cross-section  the  cross-section  of  the  excavation.  Its  area  is, 
therefore,  i  w  h.  The  height  of  this  prism  at  B  (fig.  76)  is  (§  174) 
Bi  H  —  2  H  F  sin.  D  =  2  d  sin.  D.  In  a  similar  manner,  the 
height  at  E  will  be  2  Q  E  sin.  D  =  b  sin.  D,  and  at  the  point  in- 
termediate between  (?  and  E,  the  distance  of  which  from  the  cen- 
tre line  is  \  1}  —  w,  the  height  will  be  2(%b  —  w) sin. D=  (b  —  2w) 
sin.  D.  Hence,  the  correction,  or  the  solidity  of  the  prism,  will 
be  (§  159)  C  =  i  w  h  x  i  (2  d  +  ~b  +  I  -  2  w)  sin.  D  =  |  w  h  x 
j  (d  +  i  _  w}  sjn>  2). 

When  the  excavation  lies  on  both  sides  of  the  centre  line,  the 
correction,  from  what  has  been  said  above,  is  a  triangular  prism 
having  also  for  its  cross-section  the  cross-section  of  the  excava- 
tion. Its  area  will,  therefore,  be  \  w  h.  The  height  of  this  prism 
at  B  is  also  2dsin.Z),  and  the  height  at  E,  bsin.D;  but  at  the 
point  intermediate  between  A  and  G,  the  distance  of  which  from 
the  centre  line  is  w  —  i  b,  the  height  will  be  2(w  —  $b)  sin.  D  = 
(2  w  —  b)  sin.  D.  As  this  height  is  to  be  considered  minus,  it  must 
be  subtracted  from  the  others,  and  the  correction  required  will  be 
C—\wJi  x  $(2d  +  b  —  2w  +  b)  sm.D  =  $wh  x  f  (eZ  +  b  -  w) 
sin.  D.  Hence,  in  all  cases,  when  the  section  is  partly  in  excava- 
tion and  partly  in  embankment,  we  have  the  formula 

C  =  i  w  h  x  f  (d  +  b  —  w)  sin.  D. 


When  R  is  given,  and  not  Z>,  substitute  for  sin.  D  its  value  (§  9) 

50 
sin.  D  =  —  .    The  correction  then  becomes 


. 
o  _n, 

This  correction  is  to  be  added,  when  the  highest  ground  is  on 
the  convex  side  of  the  curve,  and  subtracted  when  the  highest 
ground  is  on  the  concave  side.  At  a  tangent  point  the  correction 
will  be  just  half  of  that  given  above. 

Example.  Given  w  =  17,  b  =  30,  d  =  51,  h  =  24,  and  R  = 
1600,  to  find  C.  Here  the  area  of  the  cross-section 


NOTE   ON   THE    COMPUTATION    OF    EARTH- WORK.  163 

12  —  204,  and  one  third  of  the  sum  of  the  heights  of  the  prism  is 
100(d  +  6~w)_100(51  +  30-17)_4  _  4  _ 

3#  3x1600         ~3'  X3~ 

272  cubic  feet. 

177.  The  preceding  corrections  (§  174  and  §  176)  suppose  the 
length  of  the  sections  to  be  100  feet.  If  the  sections  are  shorter, 
the  angle  B  FH(fi.g.  75)  may  be  regarded  as  the  same  part  of  D 
that  F  G  is  of  100  feet,  and  B^FB  as  the  same  part  of  D  that 
E F  is  of  100  feet.  The  true  correction  may  then  be  taken  as  the 
same  part  of  C  that  the  sum  of  the  lengths  of  the  two  adjoining 
sections  is  of  200  feet. 


NOTE  ON  THE  COMPUTATION  OF  EARTH-WORK. 

178.  The  mode  of  computing  earth- work  on  railroads  by  first 
finding  equivalent  level- top  sections  has  already  been  examined 
in  §  172,  and  the  assumption  made  in  applying  the  prismoidal 
formula  is  shown  to  lead  to  possibly  serious  errors.  Another  as- 
sumption that  forms  the  basis  of  many  formula?,  tables,  and  dia- 
grams, is  that  the  natural  surface  of  the  ground  of  such  a  section 
as  that  calculated  in  §  166  is  a  warped  surface  or  hyperbolic  parab- 
oloid. The  solidity  is  then  computed  by  the  prismoidal  formula. 
Computing  the  section  just  referred  to  on  this  assumption,  we 
find  the  solidity  31  210  feet.  Now  we  have  seen  in  §  172  that, 
with  the  diagonal  running  in  one  direction,  the  solidity  is  32  820 
feet,  and,  with  the  diagonal  running  in  the  other  direction,  the 
solidity  is  29  600  feet.  The  assumption  of  a  warped  surface  gives, 
therefore,  an  exact  mean  between  these  two  results,  being  1,610 
feet  too  much  or  too  little,  according  to  the  direction  of  the  diag- 
onal. Errors  so  great  would  not  perhaps  be  common ;  but  they 
are  at  least  possible. 

The  objection  to  these  methods  is  that  they  involve  general  as- 
sumptions as  to  the  natural  surface  of  the  ground — assumptions 
that  the  engineer  cannot  readily  test  in  the  field  for  each  section, 
or  allow  for,  if  seen  to  be  wrong.  No  method  would  seem  to  be 
reasonably  correct  that  does  not  require  all  the  data  used  in  the 
computation  to  be  obtained  directly  in  the  field.  Now  the  division 
of  the  ground  into  triangular  planes,  whether  four  as  in  §  166,  or 
more  as  in  §  171,  satisfies  this  condition.  Since  three  points  de- 
termine a  plane,  it  is  comparatively  easy  to  decide  on  the  ground 


164  EARTH-WORK. 

what  heights  should  be  adopted  at  the  vertices,  so  that  a  triangu- 
lar plane  shall  be  a  fair  average  of  the  ground.  Suppose  the 
ground  cross-sectioned  in  the  usual  way,  and  the  actual  cats 
marked  on  the  stakes  and  recorded.  These  cuts  remain  to  guide 
the  contractor  in  his  work ;  but  the  engineer  is  to  examine  each 
triangle,  and  see  whether  these  cuts  require  any  correction  in 
order  to  obtain  a  fair  average*  of  the  surface.  As  he  goes  from 
section  to  section,  two  of  the  heights  or  cuts  would  in  general  be 
already  fixed,  and,  standing  at  the  third  vertex,  he  readily  deter- 
mines whether  the  actual  cut  there  should  stand,  or  have  one,  two, 
three,  or  more  tenths  added  or  subtracted.  The  correction,  if 
any,  may  be  noted  in  small  figures  over  the  actual  cut,  and  applied 
when  the  heights  are  taken  off  for  the  computations. 

Some  additional  labor  is  doubtless  involved  in  thus  obtaining 
directly  all  the  data  required,  and  dispensing  with  all  general  as- 
sumptions ;  but  if  justice  to  the  contractor  and  to  the  company 
require  such  additional  labor,  the  engineer  will  not  hesitate  on 
that  account.  The  computations,  as  arranged  in  §  167,  will  be 
found,  after  a  little  practice,  to  admit  of  very  rapid  work.  Of 
course,  only  final  estimates  require  so  much  care. 

In  preliminary  estimates,  where  centre  heights  alone  are  taken, 
the  method  of  §  164  will  be  found  sufficiently  accurate,  and  if  the 
computations  are  arranged  as  there  shown,  the  work  will  be  found 
very  expeditious.  In  many  cases  where  only  approximate  results 
are  aimed  at,  especially  in  making  the  usual  "  monthly  estimates," 
the  method  of  averaging  end  areas  may  be  employed.  This 
method  consists  in  finding  the  areas  of  the  two  cross-sections 
which  bound  a  section  of  an  excavation,  and  multiplying  the  aver- 
age of  these  areas  by  the  length  of  the  section  to  obtain  the  con- 
tents of  the  section. 


TABLE   I. 

RADII,    ORDINATES,    TANGENT    DEFLECTIONS,    AND 
ORDINATES  FOR  CURVING  RAILS. 

This  table  applies  directly  only  to  curves  laid  out  with  100  feet 
chords.  With  shorter  chords,  it  may  still  be  made  useful.  When 
50  feet  chords  are  used  with  a  deflection  angle  half  that  for  100 
feet  chords,  the  radius  of  the  curve  is  so  slightly  shortened,  that, 
for  the  purpose  of  finding  the  new  ordinates  and  tangent  deflec- 
tions from  Table  I.,  the  curve  is  practically  the  same  as  when 
laid  out  with  100  feet  chords.  The  change  in  the  radius  is  easily 
found.  Let  D  be  the  deflection  angle  for  100  feet  chords,  and 

XA  Kf\ 

we  have  (§  10  and  Tab.  X.,  22)  R  =  -^  =  0   .     ,  „ r-=  = 

^  sm.D      2  sin.  |  Z>  cos.  £  D 

— — ,  and  for  Ri ,  the  radius  for  50  feet  chords,  RI  = 
sin. -I- .D  cos.  \D' 

05 
.  =  R  cos. \D.    In  a  12°  curve,  where  R  =  478.34  and  D  — 

6°,  we  have  Rl  =  Rcos.3°  =  478.34  x  .99863  =  477.68.  Now  in 
the  same  curve  the  ordinates  (§  27)  and  the  tangent  deflections 
(§  19)  are  to  each  other  as  the  squares  of  the  chords;  that  is,  for 
50  feet  chords  these  quantities  are  one-fourth  of  those  given  in 
Table  I.  for  100  feet  chords.  The  ordinates  for  curving  30  feet 
rails  will,  of  course,  be  unchanged.  In  the  present  example  the 

, ,  ,     2.620  ,  1.965 

ordinates  would  be  .  =  .655  and  — - —  =  .491,  the  tangent  de- 
flection —  —  =  2.613,  and  the  ordinates  for  curving  30  feet  rails 

.235  and  .176. 

With  25  feet  chords  and  a  deflection  angle  of  1-J-0  we  should 
have  the  radius  R*  =  R  cos.  3°  cos.  1£°,  and  the  ordinates  and 
tangent  deflection  one-sixteenth  of  those  in  Table  I.,  while  the 
ordinates  for  curving  30  feet  rails  would  still  be  unchanged. 

This  curve,  strictly  speaking,  could  no  longer  be  called  a  12° 
curve.  The  new  degree,  here  about  12°  1',  might  be  found,  or 
the  curve  might  be  designated  by  the  radius ;  but  the  most  con- 
venient and  definite  designation  would  be :  Deflection  angle  3° 
for  50  feet  chords,  or  deflection  angle  1£°  for  25  feet  chords. 


TABLE  I.      RADII,    ORDINATES,    TANGENT   DEFLECTIONS, 


De- 
gree. 

Radius,  §  10. 

Ordinates,  §  25. 

Tangent 
Deflec- 
tion, §  19. 

Curving  30-ft. 
rails,  §  29. 

De- 
gree. 

m. 

fl*. 

m. 

|m. 

0    0 

Infinite. 

.000 

.000 

.000 

.000 

.000 

o       / 

0    0 

2 

171887.35 

.007 

.005 

.029 

.001 

.000 

2 

4 

85943.67 

.015 

.011 

.058 

.001 

.001 

4 

6 

57295.79 

.022 

.016 

.087 

.002 

.001 

6 

8 

42971.84 

.029 

.022 

.116 

.003 

.002 

8 

10 

34377.48 

.036 

.027 

.145 

.003 

.002 

10 

28647.91 

.044 

.033 

.175 

.004 

.003 

12 

14 

24555.35 

.051 

.038 

.204 

.005 

.003 

14 

16 

21485.94 

.058 

.044 

.233 

.005 

.004 

16 

18 

19098.62 

.065 

.049 

.262 

.006 

.004 

18 

20 

17188.76 

.073 

.055 

.291 

.007 

.005 

20 

22 

15626.15 

.080 

.060 

.320 

.007 

.005 

22 

24 

14323.97 

.087 

.065 

.349 

.008 

.006 

24 

26 

13222.13 

.095 

.071 

.378 

.009 

.006 

26 

28 

12277.70 

.102 

.076 

.407 

.009 

.007 

28 

30 

11459.19 

.109 

.082 

.436 

.010 

.007 

30 

32 

10743.00 

.116 

.087 

.465 

.010 

.008 

32 

34 

10111.06 

.124 

.093 

.495 

.011 

.008 

34 

36 

9549.34 

.131 

.098 

.524 

.012 

.009 

36 

38 

9046.75 

.138 

.104 

.553 

.012 

.009 

38 

40 

8594.41 

.145 

.109 

.582 

.013 

.010 

40 

42 

8185.16 

.153 

.115 

.611 

.014 

.010 

42 

44 

7813.11 

.160 

.120 

.640 

.014 

.011 

44 

46 

7473.42 

.167 

.125 

.669 

.015 

.011 

46 

48 

7162.03 

.175 

.131 

.698 

.016 

.012 

48 

50 

6875.55 

.182 

.136 

727 

.016 

.012 

50 

52 

6611.12 

.189 

.142 

.'756 

.017 

.013 

52 

54 

6366.26 

.196 

.147 

.785 

.018 

.013 

54 

56 

6138.90 

.204 

.153 

.814 

.018 

.014 

56 

58 

5927.22 

.211 

.158 

.844 

.019 

.014 

58 

1    0 

5729.65 

.218 

.164 

.873 

.020 

.015 

1    0 

2 

5544.83 

.225 

.169 

.902 

.020 

.015 

2 

4 

5371.56 

.233 

.175 

.931 

.021 

.016 

4 

6 

5208.79 

.240 

.180 

.960 

.022 

.016 

6 

8 

5055.59 

.247 

,185 

.989 

.022 

.017 

8 

10 

4911.15 

.255 

.191 

1.018 

.023 

.017 

10 

12 

4774.74 

.262 

.196 

1.047 

.024 

.018 

12 

14 

4645.69 

.269 

.202 

1.076 

.024 

.018 

14 

16 

4523.44 

.276 

.207 

1.105 

.025 

.019 

16 

18 

4407.46 

.284 

.213 

1.134 

.026 

.019 

18 

20 

4297.28 

.291 

.218 

1.164 

.026 

.020 

20 

22 

4192.47 

.298 

.224 

1.193 

.027 

.020 

22 

24 

4092.66 

.305 

.229 

1.222 

.027 

.021 

24 

26 

3997.48 

.313 

.235 

1.251 

.028 

.021 

26 

28 

3906.64 

.320 

.240 

1.280 

.029 

.022 

28 

30 

3819.83 

.327 

.245 

1.309 

.029 

.022 

30 

32 

3736.79 

.335 

.251 

1.338 

.030 

.023 

32 

34 

3657.29 

.342 

.256 

1.367 

.031 

.023 

34 

36 

3581.10 

.349 

.262 

1.396 

.031 

.024 

36 

38 

3508.02 

.356 

.267 

1.425 

.032 

.024 

38 

40 

3437.87 

.364 

.273 

1.454 

.033 

.025 

40 

42 

3370.46 

.371 

.278 

1.483 

.033 

.025 

42 

44 

3305.65 

.378 

.284 

1.513 

.034 

.026 

44 

46 

3243.29 

.385 

.289 

1.542 

.035 

.026 

46 

48 

3183.23 

.393 

.295 

1.571 

.035 

.026 

48 

50 

3125.36 

.400 

.300 

1.600 

.036 

.027 

50 

52 

3069.55 

.407 

.305 

1.629 

.037 

.027 

52 

54 

3015.71 

.415 

.311 

1.658 

.037 

.028 

54 

56 

2963.72 

.422 

.316 

1.687   . 

.038 

.028 

56 

58 

2913.49 

.429 

.322 

1.716 

.039 

.029 

58 

AND   OBDINATES   FOB   CURVING   BAILS. 


De- 
gree. 

Radius,  §10. 

Ordinates,  §  25. 

Tangent 
Deflec- 
tion, §19 

Curving  30-ft. 
rails,  §  29. 

De- 
gree. 

m. 

£  m. 

m. 

f  m. 

0          / 

2    0 

2864.93 

.436 

.327 

1.745 

.039 

.029 

0         / 

2    0 

2 

2817.97 

.444 

.333 

1.774 

.040 

.030 

2 

4 

2772.53 

.451 

.338 

1.803 

.041 

.030 

4 

6 

2728.52 

.458 

.344 

1.832 

.041 

.031 

6 

8 

2685.90 

.465 

.349 

1.862 

.042 

.031 

8 

10 

2644.58 

.473 

.355 

1.891 

.043 

.032 

10 

12 

2604.51 

.480 

.360 

1.920 

.043 

.032 

12 

14 

2565.65 

.487 

.365 

1.949 

.044 

.033 

14 

16 

2527.92 

.495 

.371 

1.978 

.045 

.033 

16 

18 

2491.29 

.502 

.376 

2.007 

.045 

.034 

18 

20 

2455.70 

.509 

.382 

2.036 

.046 

.034 

20 

22 

2421.12 

.516 

.387 

2.065 

.046 

.035 

22 

24 

2387.50 

.524 

393 

2.094 

.047 

.035 

24 

26 

2354.80 

.531 

'.398 

2.123 

.048 

.036 

26 

28 

2322.98 

.538 

.404 

2.152 

.048 

.036 

28 

30 

2292.01 

.545 

.409 

2.181 

.049 

.037 

30 

32 

2261.86 

.553 

.415 

2.211 

.050 

.037 

32 

34 

2232.49 

.560 

.420 

2.240 

.050 

.038 

34 

36 

2203.87 

.567 

.425 

2.269 

.051 

.038 

36 

•  38 

2175.98 

.575 

.431 

2.298 

.052 

.039 

38 

40 

2148  79 

.582 

.436 

2.327 

.052 

.039 

40 

42 

2122.26 

.589 

.442 

2.356 

.053 

.040 

42 

44 

2096.39 

.596 

.447 

2.385 

.054 

.040 

44 

46 

2071.13 

.604 

.453 

2.414 

.054 

.041 

46 

48 

2046.48 

.611 

.458 

2.443 

.055 

.041 

48 

50 

2022.41 

.618 

.464 

2.472 

.056 

.042 

50 

52 

1998.90 

.625 

.469 

2.501 

.056 

.042 

52 

54 

1975.93 

.633 

.475 

2.530 

.057 

.043 

54 

56 

1953.48 

.640 

.480 

2.560 

.058 

.043 

56 

58 

1931.53 

.647 

.485 

2.589 

.058 

.044 

58 

3    0 

1910.08 

.655 

.491 

2.618 

.059 

.044 

3    0 

2 

1889.09 

.662 

.496 

2.647 

.060 

.045 

2 

4 

1868.56 

.669 

.502 

2.676 

.060 

.045 

4 

6 

1848.48 

.676 

.507 

2.705 

.061 

.046 

6 

8 

1828.82 

.684 

.513 

2.734 

.062 

.046 

8 

10 

1809.57 

.691 

.518 

2.763 

.062 

.047 

10 

12 

1790.73 

.698 

.524 

2.792 

.063 

.047 

12 

14 

1772.27 

.705 

.529 

2.821 

.063 

.048 

14 

16 

1754.19 

.713 

.535 

2.850 

.064 

.048 

16 

18 

1736.48 

.720 

.540 

2.879 

.065 

.049 

18 

20 

1719.12 

.727 

.545 

2.908 

.065 

.049 

20 

22 

1702.10 

.735 

.551 

2.938 

.066 

.050 

22 

24 

1685.42 

.742 

.556 

2.967 

.067 

.050 

24 

26 

1669.06 

.749 

.562- 

2.996 

.067 

.051 

26 

28 

1653.01 

.756 

.567 

3.025 

.068 

.051 

28 

30 

1637.28 

.764 

.573 

3.054 

.069 

.052 

30 

32 

1621.84 

.771 

.578 

3.083 

.069 

.052 

32 

34 

1606.68 

.778 

.584 

3.112 

.070 

.053 

34 

36 

1591.81 

.785 

.589 

3.141 

.071 

.053 

36 

38 

1577.21 

.793 

.595 

3.170 

.071 

.053 

38 

40 

1562.88 

.800 

.600 

3.199 

.072 

.054 

40 

42 

1548.80 

.807 

.605 

3.228 

.073 

.054 

42 

44 

1534.98 

.815 

.611 

3.257 

.073 

.055 

44 

46 

1521.40 

.822 

.616 

3.286 

.074 

.055 

46 

48 

1508.06 

.829 

.622 

3.316 

.075 

.056 

48 

50 

1494.95 

.836 

.627 

3.345 

.07'5 

.056 

50 

52 

1482.07 

.844 

.633 

3.374 

.076 

.057 

52 

54 

1469.41 

.1851 

.638 

3.403 

.077 

.057 

54 

56 

1456.96 

.858 

.644 

3.432 

.077 

.058 

56 

58 

1444.72 

.865 

.649 

3.461 

.078 

.058 

58 

168     TABLE    I.       RADII,    ORDINATES,    TANGENT   DEFLECTIONS, 


De- 
gree. 

Radius,  §10. 

Ordinates,  §  25. 

Tangent 
Deflec- 
ti.n,§19. 

Curving  30-ft. 
rails,  §  29. 

De- 
gree. 

m. 

Im 

m. 

fm. 

4    0 

1432.69 

.873 

.655 

3.490 

.079 

.059 

4    0 

2 

1420.85 

.880 

.660 

3.519 

.079 

.059 

2 

4 

1409.21 

.887 

.665 

3.548 

.080 

.060 

4 

6 

1397.76 

.895 

.671 

3.577 

.080 

.060 

6 

8 

1386.49 

.902 

.676 

3.606 

.081 

.061 

8 

10 

1375.40 

.909 

.682 

3.635 

.082 

.061 

10 

12 

1364.49 

.916 

.687 

3.664 

.082 

.062 

12 

14 

1353.75 

.924 

.693 

3.693 

.083 

.062 

14 

16 

1341118 

.931 

.698 

3.723 

.084 

.063 

16 

18 

1332.77 

.938 

.704 

3.752 

.084 

.063 

18 

20 

1322.53 

.946 

.709 

3.781 

.085 

.064 

20 

22 

1312.43 

.953 

.715 

3.810 

.086 

.064 

22 

24 

1302.50 

.960 

.720 

3.839 

.086 

.065 

24 

26 

1292.71 

.967 

.725 

3.868 

.087 

.065 

26 

28 

1283.07 

.975 

.731 

3.897 

.088 

.066 

28 

30 

1273.57 

.982 

.736 

3.926 

.088 

.066 

30 

32 

1264.21 

.989 

.742 

3.955 

.089 

.067 

32 

34 

1254.98 

.996 

.747 

3.984 

.090 

.067 

34 

36 

1245.89 

1.004 

.753 

4.013 

.090 

.068 

36 

38 

1236.94 

1.011 

.758 

4.042 

.091 

.068 

-38 

40 

1228.11 

1.018 

.764 

4.071 

.092 

.069 

40 

42 

1219.40 

1.026 

.769 

4.100 

.092 

.069 

42 

44 

1210.82 

1.033 

.775 

4.129 

.093 

.070 

44 

46 

1202.36 

1.040 

.780 

4.159 

.094 

.070 

46 

48 

1194.01 

1.047 

.786 

4.188 

.094 

.071 

48 

50 

1185.78 

1.055 

.791 

4.217 

.095 

.071 

50 

52 

1177.66 

1.062 

.796 

4.246 

.096 

.072 

52 

54 

1169.66 

1.069 

.802 

4.275 

.096 

.072 

64 

56 

1161.76 

1.076 

.807 

4.304 

.097 

.073 

56 

58 

1153.97 

1.084 

.813 

4.333 

.097 

.073 

58 

5    0 

1146.28 

1.091 

.818 

4.362 

.098 

.074 

5    0 

2 

1138.69 

1.098 

.824 

4.391 

.099 

.074 

2 

4 

1131.21 

1.106 

.829 

4.420 

.099 

.075 

4 

6 

1123.82 

1.113 

.835 

4.449 

.100 

.075 

6 

8 

1116.52 

1.120 

.840 

4.478 

.101 

.076 

8 

10 

1109.33 

1.127 

.846 

4.507 

.101 

.076 

10 

12 

1102.22 

1.135 

.851 

4.536 

.102 

.077 

12 

14 

1095.20 

1.142 

.856 

4.565 

.103 

.077 

14 

16 

1088.28 

1.149 

.862 

4.594 

.103 

.078 

16 

18 

1081.44 

1.156 

.867 

4.623 

.104 

.078 

18 

20 

1074.68 

1.164 

.873 

4.653 

.105 

.079 

20 

22 

1068.01 

1.171 

.878 

4.682 

.105 

.079 

22 

24 

1061.43 

1.178 

.884 

4.711 

.106 

.079 

24 

26 

1054.92 

1.186 

.889 

4.740 

.107 

.080 

26 

28 

1048.49 

1.193 

.895 

4.769 

.107 

.080 

28 

30 

1042.14 

1.200 

.900 

4.798 

.108 

.081 

30 

32 

1035.87 

1.207 

.906 

4.827 

.109 

.081 

32 

34 

1029.67 

1.215 

.911 

4.856 

.109 

.082 

34 

36 

1023.55 

1.222 

.916 

4.885 

.110 

.082 

36 

38 

1017.49 

1.229 

.922 

4.914 

.111 

.083 

38 

40 

1011.51 

1.237 

.927 

4.943 

.111 

.083 

40 

42 

1005.60 

1.244 

.933 

4.972 

.112 

.084 

42 

44 

999.76 

1.251 

.938 

5.001 

.113 

.084 

44 

46 

993.99 

1.258 

.944 

5.030 

.113 

.085 

46 

48 

988.28 

1.266 

.949 

5.059 

.114 

.085 

48 

50 

982.64 

1.273 

.955 

5.088 

.114 

.086 

50 

52 

977.06 

1.280 

.960 

5.117 

.115 

.086 

52 

54 

971.54 

1.287 

.966 

5.146 

.116 

.087 

54 

56 

966.09 

1.295 

.971 

5.175 

.116 

.087 

56 

58 

960.70 

1.302 

.977 

5.205 

.117 

.088 

58 

AND   OBDINATES  FOB   CUBVING   BAILS. 


De- 
gree. 

Radius,  §10. 

Ordinates,  §  25. 

Tangent 
Deflec- 
tion, §19. 

Curving  30-ft. 
rails,  §  29. 

De- 
gree. 

m. 

im. 

m. 

f  m. 

6    0 

955.37 

1.309 

.982 

5.234 

.118 

.088 

0         / 

6    0 

2 

950.09 

1.317 

.987 

5.263 

.118 

.089 

2 

4 

944.88 

1.324 

.993 

5.292 

.119 

.089 

4 

6 

939.72 

1.331 

.998 

5.321 

.120 

.090 

6 

8 

934.62 

1.338 

1.004 

5.350 

.120 

.090 

8 

10 

929.57 

1.346 

1.009 

5.379 

.121 

.091 

10 

12 

92458 

1.353 

1.015 

5.408 

.122 

.091 

12 

14 

919.64 

1.360 

1.020 

5.437 

.122 

.092 

14 

16 

914.75 

1.368 

1.026 

5.466 

.123 

.092 

16 

18 

909.92 

1.375 

1.031 

5.495 

.124 

.093 

18 

20 

905.13 

1.382 

1.037 

5.524 

.124 

.093 

20 

22 

900.40 

1.389 

1.042 

5.553 

.125 

.094 

22 

24 

895.71 

1.397 

1.047 

5.582 

.126 

.094 

24 

26 

891.08 

1.404 

1.053 

5.611 

.126 

.095 

26 

28 

886.49 

1.411 

1.058 

5.640 

.127 

.095 

28 

30 

881.95 

1.418 

1.064 

5.669 

.128 

.096 

30 

32 

877.45 

1.426 

1.069 

5.698 

.128 

.096 

32 

34 

873.00 

1.433 

1.075 

5.727 

.129 

.097 

34 

36 

868.60 

1.440 

1.080 

5.756 

.130 

.097 

36 

38 

864.24 

1.448 

1.086 

5.785 

.130 

.098 

38 

40 

859.92 

1.455 

1.091 

5.814 

.131 

.098 

40 

42 

855.65 

1.462 

1.097 

5.844 

.131 

.099 

42 

44 

851.42 

1.469 

1.102 

5.873 

.132 

.099 

44 

46 

847.23 

1.477 

1.108 

5.902 

.133 

.100 

46 

48 

843.08 

1.484 

1.113 

5.931 

.133 

.100 

48 

50 

838.97 

1.491 

1.118 

5.960 

.134 

.101 

50 

52 

834.90 

1.499 

1.124 

5.989 

.135 

.101 

52 

54 

830.88 

1.506 

1.129 

6.018 

.135 

.102 

54 

56 

826.89 

1.513 

1.135 

6.047 

.136 

.102 

56 

58 

822.93 

1.520 

1.140 

6.076 

.137 

.103 

58 

7    0 

819.02 

1.528 

1.146 

6.105 

.137 

.103 

7    0 

2 

815.14 

1.535 

.151 

6.134 

.138 

.104 

2 

4 

811.30 

1.542 

.157 

6.163 

.139 

.104 

4 

6 

807.50 

1.549 

.162 

6.192 

.139 

.104 

6 

8 

803.73 

1.557 

.168 

6.221 

.140 

.105 

8 

10 

800.00 

1.564 

.173 

6.250 

.141 

.105 

10 

12 

796.30 

1.571 

.178 

6.279 

.141 

.106 

12 

14 

792.63 

1.579 

.184 

6.308 

.142 

.106 

14 

16 

789.00 

1.586 

'1.189 

6.337 

.143 

.107 

16 

18 

785.40 

1.593 

1.195 

6.366 

.143 

.107 

18 

20 

781.84 

1.600 

1.200 

6.395 

.144 

.108 

20 

22 

778.31 

1.608 

1.206 

6.424 

.145 

.108 

22 

24 

774.81 

1.615 

1.211 

6.453 

.145 

.109 

24 

26 

771.34 

1.622 

1.217 

6.482 

.146 

.109 

26 

28 

767.90 

1.630 

1.222 

6.511 

.147 

.110 

28 

30 

764.49 

1.637 

1.228 

6.540 

.147 

.110 

30 

32 

761.11 

1.644 

1.233 

6.569 

.148 

.111 

32 

34 

757.76 

1.651 

1.239 

6.598 

.148 

.111 

34 

36 

754.44 

1.659 

1.244 

6.627 

.149 

.112 

36 

38 

751.16 

1.666 

1.249 

6.656 

.150 

.112 

38 

40 

747.89 

1.673 

1.255 

6.685 

.150 

.113 

40 

42 

744.66 

1.681 

1.260 

6.714 

.151 

.113 

42 

44 

741.46 

1.688 

1.266 

6.743 

.152 

.114 

44 

46 

738.28 

1.695 

1.271 

6.773 

.152 

.114 

46 

48 

735.13 

1.702 

1.277 

6.802 

.153 

.115 

48 

50 

732.01 

1.710 

1.282 

6.831 

.154 

.115 

50 

52 

728.91 

1.717 

1.288 

6.860 

.154 

.116 

52 

54 

725.84 

1.724 

1.293 

6.889 

.155 

.116 

54 

56 

722.79 

1.731 

1.299 

6.918 

.156 

.117 

56 

58 

719.77 

1.739 

1.304 

6.947 

.156 

.117 

58 

170     TABLE  I.      BADII,    ORDINATES,    TANGENT   DEFLECTIONS, 


De- 
gree. 

Radius,  §10. 

Ordinates,  §  25. 

Tangent 
Deflec- 
tion, §19. 

Curving  30-ft. 
rails,  §  29. 

De- 
gree. 

m. 

1-  m. 

m. 

f  m. 

0          / 

8    0 

71678 

1.746 

1.310 

6.976 

.157 

.118 

8    0 

2 

713.81 

1.753 

1.315 

7.005 

.158 

.118 

2 

4 

710.87 

1.761 

1.320 

7.034 

.158 

.119 

4 

6 

707.94 

1.768 

1.326 

7.063 

.159 

.119 

6 

8 

705.05 

1.775 

1.331 

7.093 

.160 

.120 

8 

10 

702.18 

1.782 

1.337 

7.121 

.160 

.120 

10 

12 

699.33 

1.790 

1.342 

7.150 

.161 

.121 

12 

14 

696.50 

1.797 

1.348 

7.179 

.162 

.121 

14 

16 

693.70 

1.804 

1.353 

7.208 

.162 

.122 

16 

18 

690.91 

1.812 

1.359 

7.237 

.163 

.122 

18 

20 

688.16 

1.819 

1.364 

7.266 

.163 

.123 

20 

22 

685.42 

1.826 

1.370 

7.295 

.164 

.123 

22 

24 

683.70 

1.833 

1.375 

7.324 

.165 

.124 

24 

26 

680.01 

1.841 

1.381 

7.353 

.165 

.124 

26 

28 

677.34 

1.848 

1.386 

7.382 

.166 

.125 

28 

30 

674.69 

1.855 

1.391 

7.411 

.167 

.125 

30 

32 

672.06 

1.863 

1.397 

7.440 

.167 

.126 

32 

34 

669.45 

1.870 

1.402 

7.469 

.168 

.126 

34 

36 

666.86 

1.877 

1.408 

7.498 

.169 

.127 

36 

38 

664.29 

1.884 

1.413 

7.527 

.169 

.127 

38 

40 

661.74 

1.892 

1.419 

7.556 

.170 

.128 

40 

42 

659.21 

1.899 

1.424 

7.585 

.171 

.128 

42 

44 

656.69 

1.906 

1.430 

7.614 

.171 

.128 

44 

46 

654.20 

1.914 

1.435 

7.643 

.172 

.129 

46 

48 

651.73 

1.921 

1.441 

7.672 

.173 

.129 

48 

50 

649.27 

1.928 

1.446 

7.701 

.173 

.130 

50 

52 

64684 

1.935 

1.452 

7.730 

.174 

.130 

52 

54 

644.42 

1.943 

1.457 

7.759 

.175 

.131 

54 

56 

642.02 

1.950 

1.462 

7.788 

.175 

.131 

56 

58 

639.64 

1.957 

1.463 

7.817 

.176 

.132 

58 

9    0 

637.27 

1.965 

1.473 

7.846 

.177 

.132 

9    0 

2 

634.93 

1.972 

1.479 

7.875 

.177 

.133 

2 

4 

632.60 

1.979 

1.484 

7.904 

.178 

.133 

4 

6 

630.29 

1.986 

1.490 

7.933 

.178 

.134 

6 

8 

627.99 

1.994 

1.495 

7.962 

.179 

.134 

8 

10 

625.71 

2.001 

1.501 

7.991 

.180 

.135 

10 

12 

623.45 

2.008 

1.506 

8.020 

.180 

.135 

12 

14 

621.20 

2.015 

1.512 

8.049 

.181 

.136 

14 

16 

618.97 

2.023 

1.517 

8.078 

.182 

.136 

16 

18 

616.76 

2.030 

1.523 

8.107 

.182 

.137 

18 

20 

614.56 

2.037 

1.528 

8.136 

.183 

.137 

20 

22 

612.38 

2.045 

1.533 

8.165 

.184 

.138 

22 

24 

610.21 

2.052 

1.539 

8.194 

.184 

.138 

24 

26 

608.06 

2.059 

1.544 

8.223 

.185 

.139 

26 

28 

605.93 

2.066 

1.550 

8.252 

.186 

.139 

28 

30 

603.80 

2.074 

1.555 

8.281 

.186 

.140 

30 

32 

601.70 

2.081 

1.561 

8.310 

.187 

.140 

32 

34 

599.61 

2.088 

1.566 

8.339 

.188 

.141 

34 

36 

597.53 

2096 

1.572 

8.368 

.188 

.141 

36 

38 

595.47 

2.103 

1.577 

8.397 

.189 

.142 

38 

40 

593.42 

2.110 

1.583 

8.426 

.190 

.142 

40 

42 

591.38 

2.117 

1.588 

8.455 

.190 

.143 

42 

44 

589.36 

2.125 

1.594 

8.484 

.191 

.143 

44 

46 

587.36 

2.132 

1.599 

8.513 

.192 

.144 

46 

48 

585.36 

2.139 

1.604 

8.542 

.192 

.144 

48 

50 

583.38 

2.147 

1.610 

8.571 

.193 

.145 

50 

52 

581.42 

2.154 

1.615 

8.600 

.193 

.145 

52 

54 

579.47 

2.161 

1.621 

8.629 

.194 

.146 

54 

56 

577.53 

2.168 

1.626 

8.658 

.195 

.146 

56 

58 

575.60 

2.176 

1.632 

8.687 

.195 

.147 

58 

AND   ORDINATE8   FOB   CURVING   RAILS. 


171 


De- 
gree. 

Radius,  §  10. 

Ordinates,  §  25. 

Tangent 
Deflec- 
tion,!^. 

Curving  30-f  t. 
rails,  §  29. 

De- 
gree. 

m. 

fm. 

m. 

|  m. 

o        / 

10    0 

573.69 

2.183 

1.637 

8.716 

.196 

.147 

10    0 

4 

569.90 

2.198 

1.648 

8.774 

.197 

.148 

4 

8 

566.16 

2.212 

1.659 

8.831 

.199 

.149 

8 

12 

562.47 

2.227 

1.670 

8.889 

.200 

.150 

12 

16 

558.82 

2.241 

1.681 

8.947 

.201 

.151 

16 

20 

555.23 

2.256 

1.692 

9.005 

.203 

.152 

20 

24 

551.68 

2.270 

1.703 

9.063 

.204 

.153 

24 

28 

548.17 

2.285 

1.714 

9.121 

.205 

.154 

28 

32 

544.71 

2.300 

1.725 

9.179 

.207 

.155 

32 

36 

541.30 

2.314 

1.736 

9.237 

.208 

.156 

36 

40 

537.92 

2.329 

1.747 

9.295 

.209 

.157 

40 

44 

534.59 

2.343 

1.758 

9.353 

.210 

.158 

44 

48 

531.30 

2.358 

1.768 

9.411 

.212 

.159 

48 

52 

528.05 

2.373 

1.779 

9.469 

.213 

.160 

52 

56 

524.84 

2.387 

1.790 

9.527 

.214 

.161 

56 

11    0 

521.67 

2.402 

1.801 

9.585 

.216 

.162 

11    0 

4 

518.54 

2.416 

1.812 

9.642 

.217 

.163 

4 

8 

515.44 

2.431 

1.823 

9.700 

.218 

.164 

8 

12 

512.38 

2.445 

1.834 

9.758 

.220 

.165 

12 

16 

509.36 

2.460 

1.845 

9.816 

.221 

.166 

16 

20 

506.38 

2.475 

1.856 

9.874 

.222 

.167 

20 

24 

503.42 

2.489 

1.867 

9.932 

.223 

.168 

24 

28 

500.51 

2.504 

1.878 

9.990 

.225 

.169 

28 

32 

497.62 

2.518 

1.889 

10.048 

.226 

.170 

32 

36 

494.77 

2.533 

1.900 

10.106 

.227 

.171 

36 

40 

491.96 

2.547 

1.911 

10.164 

.229 

.172 

40 

44 

489.17 

2.562 

1.922 

10.221 

.230 

.172 

44 

48 

486.42 

2.577 

1.932 

10.279 

.231 

.173 

48 

52 

483.69 

2.591 

1.943 

10.337 

.233 

.174 

52 

56 

481.00 

2.606 

1.954 

10.395 

.234 

.175 

56 

12    0 

478.34 

2.620 

1.965 

10.453 

.235 

.176 

12    0 

4 

475.71 

2.635 

1.976 

10.511 

.236 

.177 

4 

8 

473.10 

2.650 

1.987 

10.569 

.238 

.178 

8 

12 

470.53 

2.664 

1.998 

10.626 

.239 

.179 

12 

16 

467.98 

2.679 

2.009 

10.684 

.240 

.180 

16 

20 

465.46 

2.693 

2.020 

10.742 

.242 

.181 

20 

24 

462.97 

2.708 

2.031 

10.800 

.243 

.182 

24 

28 

460.50 

2.722 

2.042 

10.858 

.244 

.183 

28 

32 

458.06 

2.737 

2.053 

10.916 

.246 

.184 

32 

36 

455.65 

2.752 

2.064 

10.973 

.247 

.185 

36 

40 

453.26 

2.766 

2.075 

11.031 

.248 

.186 

40 

44 

450.89 

2.781 

2.086 

11.089 

.250 

.187 

44 

48 

448.56 

2.795 

2.097 

11.147 

.251 

.188 

48 

52 

446.24 

2.810 

2.108 

11.205 

.252 

.189 

52 

56 

443.95 

2.825 

2.118 

11.263 

.253 

.190 

56 

13    0 

441.68 

2.839 

2.129 

11.320 

.255 

.191 

13    0 

4 

439.44 

2.854 

2.140 

11.378 

.256 

.192 

4 

8 

437.22 

2.868 

2.151 

11.436 

.257 

.193 

8 

12 

435.02 

2.883 

2.162 

11.494 

.259 

.194 

12 

16 

432.84 

2.898 

2.173 

11.552 

.260 

.195 

16 

20 

430.69 

2.912 

2.184 

11.609 

.261 

.196 

20 

24 

428.56 

2.927 

2.195 

11.667 

.263 

.197 

24 

28 

426.44 

2.941 

2.206 

11.725 

.264 

.198 

28 

32 

424.35 

2.956 

2.217 

11.783 

.265 

.199 

32 

36 

422.28 

2.971 

2.228 

11.840 

.266 

.200 

36 

40 

420.23 

2.985 

2.239 

11.898 

.268 

.201 

40 

44 

418.20 

3.000 

2.250 

11.956 

.269 

.202 

44 

48 

416.19 

3.014 

2.261 

12.014 

.270 

.203 

48 

52 

414.20 

3.029 

2.272 

12.071 

.272 

.204 

52 

56 

412.23 

3.044 

2.283 

12.129 

.273 

.205 

56 

172     TABLE   I.      RADII,    ORDINATES,    TANGENT   DEFLECTIONS, 


De- 
gree. 

Radius,  §10. 

Ordinates,  §  25. 

Tangent 
Deflec- 
tion, §19. 

Curving  30-ft. 
rails,  §  29. 

De- 
gree. 

m. 

*m. 

m. 

lift. 

0           / 

14    0 

410.28 

3.058 

2.294 

12.187 

.274 

.206 

o        / 

14    0 

4 

408.34 

3.073 

2.305 

12.245 

.276 

.207 

4 

8 

406.42 

3.087 

2.316 

12.302 

.277 

.208 

8 

12 

404.53 

3.102 

2.326 

12.360 

.278 

.209 

12 

16 

402.65 

3.117 

2.337 

12.418 

.279 

.210 

16 

20 

400.78 

3.131 

2.348 

12.476 

.281 

.211 

20 

24 

398.94 

3.146 

2.359 

12.533 

.282 

.211 

24 

28 

397.11 

3.160 

2.370 

12.591 

.283 

.212 

28 

32 

395.30 

3.175 

2.381 

12.649 

.285 

.213 

32 

36 

393.50 

3.190 

2.392 

12.706 

.286 

.214 

36 

40 

391.72 

3.204 

2.403 

12.764 

.287 

.215 

40 

44 

389.96 

3.219 

2.414 

12.822 

.288 

.216 

44 

48 

388.21 

3.233 

2.425 

12.880 

.290 

.217 

48 

52 

386.48 

3.248 

2.436 

12.937 

.291 

.218 

52 

56 

384.77 

3.263 

2.447 

12.995 

.292 

.219 

56 

15    0 

383.06 

3.277 

2.458 

13.053 

.294 

.220 

15    0 

4 

381.38 

3.292 

2.469 

13.110 

.295 

.221 

4 

8 

379.71 

3.306 

2.480 

13.168 

.296 

.222 

8 

12 

378.05 

3.321 

2.491 

13.226 

.298 

.223 

12 

16 

376.41 

3.336 

2.502 

13.283 

.299 

.224 

16 

20 

374.79 

3.350 

2.513 

13.341 

.300 

.225 

20 

24 

373.17 

3.365 

2.524 

13.399 

.301 

.226 

24 

28 

371.57 

3.379 

2.535 

13.456 

.303 

.227 

28 

32 

369.99 

3.394 

2.546 

13.514 

.304 

.228 

32 

36 

368.42 

3.409 

2.556 

13.572 

.305 

.229 

36 

40 

366.86 

3.423 

2.567 

13.629 

.307 

.230 

40 

44 

365.31 

3.438 

2.578 

13.687 

.308 

.231 

44 

48 

363.78 

3.452 

2.589 

13.744 

.309 

.232 

48 

52 

362.26 

3.467 

2.600 

13.802 

.311 

.233 

52 

56 

360.76 

3.482 

2.611 

13.860 

.312 

.234 

56 

16    0 

359.26 

3.496 

2.622 

13.917 

.313 

.235 

16    0 

4 

357.78 

3.511 

2.633 

13.975 

.314 

.236 

4 

8 

356.32 

3.526 

2.644 

14.033 

.316 

.237 

8 

12 

354.86 

3.540 

2.655 

14.090 

.317 

.238 

12 

16 

353.41 

3.555 

2.666 

14.148 

.318 

.239 

16 

20 

351.98 

3.569 

2.677 

14.205 

.320 

.240 

20 

24 

350.06 

3.584 

2.688 

14.263 

.321 

.241 

24 

28 

349.15 

3.599 

2.699 

14.320 

.322 

.242 

28 

32 

347.75 

3.613 

2.710 

14.378 

.324 

.243 

32 

36 

346.37 

3.628 

2.721 

14.436 

.325 

.244 

36 

40 

344.99 

3.643 

2.732 

14.493 

.326 

.245 

40 

44 

343.62 

3.657 

2.743 

14.551 

.327 

.246 

44 

48 

342.27 

3.672 

2.754 

14.608 

.329 

.247 

48 

52 

340.93 

3.686 

2.765 

14.666 

.330 

.247 

52 

56 

339.60 

3.701 

2.776 

14.723 

.331 

.248 

56 

17    0 

338.27 

3.716 

2.787 

14.781 

.333 

.249 

17    0 

4 

336.96 

3.730 

2.798 

14.838 

.334 

.250 

4 

8 

335.66 

3.745 

2.809 

14.896 

.335 

.251 

8 

12 

334.37 

3.760 

2.820 

14.954 

.336 

.252 

12 

16 

333.09 

3.774 

2.831 

15.011 

.338 

.253 

16 

20 

331.82 

3.789 

2.842 

15.069 

.339 

.254 

20 

24 

330.55 

3.803 

2.853 

15.126 

.340 

.255 

24 

28 

329.30 

3.818 

2.864 

15.184 

.342 

.256 

28 

32 

328.06 

3.833 

2.875 

15.241 

.343 

.257 

32 

36 

326.83 

3.847 

2.885 

15.299 

.344 

.258 

36 

40 

325.60 

3.862 

2.896 

15.356 

.346 

.259 

40 

44 

324.39 

3.877 

2.907 

15.414 

.347 

.260 

44 

48 

323.18 

3.891 

2.918 

15.471 

.348 

.261 

48 

52 

321.99 

3.906 

2.929 

15.529 

.349 

.262 

52 

56 

320.80 

3.920 

2.940 

15.586 

.351 

.263 

56 

AND    ORDINATES   FOR   CURVING   RAILS. 


De- 
gree. 

Radius,  §10. 

Ordinates,  §  25. 

Tangent 
Deflec- 
tion, §19. 

Curving  30-ft. 
rails,  §  29. 

De- 
gree. 

m. 

f  m. 

m. 

im. 

18    0 

319.62 

3.935 

2.951 

15.643 

.352 

.264 

18    0 

4 

318.45 

3.950 

2.962 

15.701 

.353 

.265 

4 

8 

317.29 

3.964 

2.973 

15.758 

.355 

.266 

8 

12 

316.14 

3.979 

2.984 

15.816 

.356 

.267 

12 

16 

315.00 

3.994 

2.995 

15.873 

.357 

.268 

16 

20 

313.86 

4.008 

3.006 

15.931 

.358 

.269 

20 

24 

312.73 

4.023 

3.017 

15.988 

.360 

.270 

24 

28 

311.61 

4.038 

3.028 

16.046 

.361 

.271 

28 

32 

310.50 

4.052 

3.039 

16.103 

.362 

.272 

32 

36 

309.40 

4.067 

3.050 

16.160 

.364 

.273 

36 

40 

308.30 

4.081 

3.061 

16.218 

.365 

.274 

40 

44 

307.22 

4.096 

3.072 

16.275 

.366 

.275 

44 

48 

306.14 

4.111 

3.083 

16.333 

.367 

.276 

48 

52 

305.06 

4.125 

3.094 

16.390 

.369 

.277 

52 

56 

304.00 

4.140 

3.105 

16.447 

.370 

.278 

56 

19    0 

302.94 

4.155 

3.116 

16.505 

.371 

.279 

19    0 

4 

301.89 

4.169 

3.127 

16.562 

.373 

.279 

4 

8 

300.85 

4.184 

3.138 

16.620 

.374 

.280 

8 

12 

299.82 

4.199 

3.149 

16.677 

.375 

.281 

12 

16 

298.79. 

4.213 

3.160 

16.734 

.377 

.282 

16 

20 

297.77 

4.228 

3.171 

16.792 

.378 

.283 

20 

24 

296.75 

4.243 

3.182 

16.849 

.379 

.284 

24 

28 

295.75 

4.257 

3.193 

16.906 

.380 

.285 

28 

32 

294.75 

4.272 

3.204 

16.964 

.382 

.286 

32 

36 

293.76 

4.287 

3.215 

17.021 

.383 

.287 

36 

40 

292.77 

4.301 

3.226 

17.078 

.384 

.288 

40 

44 

291.79 

4.316 

3.237 

17.136 

.386 

.289 

44 

48 

290.82 

4.aso 

3.248 

17.193 

.387 

.290 

48 

52 

289.85 

4.345 

3.259 

17.250 

.388 

.291 

52 

56 

288.89 

4.360 

3.270 

17.308 

.389 

.292 

56 

20    0 

287.94 

4.374 

3.281 

17.365 

.391 

.293 

20    0 

10 

285.58 

4.411 

3.308 

17.508 

.394 

.295 

10 

20 

288.27 

4.448 

3.336 

17.651 

.397 

.298 

20 

30 

280.99 

4.484 

3.363 

17.794 

.400 

.300 

30 

40 

278.75 

4.521 

3.391 

17.937 

.404 

.303 

40 

50 

276.54 

4.558 

3.418 

18.081 

.407 

.305 

50 

21    0 

274.37 

4.594 

3.446 

18.224 

.410 

.308 

21    0 

10 

272.23 

4.631 

3.473 

18.367 

.413 

.310 

10 

20 

270.13 

4.668 

3.501 

18.509 

.416 

.312 

20 

30 

268.06 

4.704 

3.528 

18.652 

.420 

.315 

30 

40 

266.02 

4.741 

3.556 

18.795 

.423 

.317 

40 

50 

264.02 

4.778 

3.583 

18.938 

.426 

.320 

50 

22    0 

262.04 

4.814 

3.611 

19.081 

.429 

.322 

22    0 

10 

260.10 

4.851 

3.638 

19.224 

.433 

.324 

10 

20 

258.18 

4.888 

3.666 

19.366 

.436 

.327 

20 

30 

256.29 

4.925 

3.693 

19.509 

.439 

.329 

30 

»^feL«  J&4.43 

4.961 

3.721 

19.652 

.442 

.332 

40 

>/#•     •K2.60 

4.998 

3.749 

19.794 

.445 

.334 

50 

23    OCTJP  250.79 

5.035 

3.776 

19.937 

.449 

.336 

23    0 

10           249.01 

5.071 

3.804 

20.079 

.452 

.339 

10 

20 

247.26 

5.108 

3.831 

20.222 

.455 

.341 

20 

30 

245.53 

5.145 

3.859 

20.364 

.458 

.344 

30 

40 

243.82 

5.182 

3.886 

20.507 

.461 

.346 

40 

50 

242.14 

5.218 

3.914 

20.649 

.465 

.348 

50 

24    0 

240.49 

5.255 

3.941 

20.791 

.468 

.351 

24    0 

10 

238.85 

5.292 

3.969 

20.933 

.471 

.353 

10 

20 

237.24 

5.329 

3.997 

21.076 

.474 

.356 

20 

30 

235.65 

5.366 

4.024 

21.218 

.477 

.358 

30 

40 

234.08 

5.402 

4.052 

21.360 

.481 

.360 

40 

50 

232.54 

5.439 

4.079 

21.502 

.484 

.363 

50 

174 


TABLE  II.       LONG    CHORDS. 

TABLE  II. 

LONG  CHORDS.    §83. 


Degree 
of  Curve. 

2  Stations. 

3  Stations. 

4  Stations. 

5  Stations. 

6  Stations. 

0  10 

200.000 

299.999 

399.998 

499.996 

599.993 

20 

199.999 

299.997 

399.992 

499.983 

599.970 

30 

199.998 

299.992 

399.981 

499.962 

599.933 

40 

199.997 

299.986 

399.966 

499.932 

599.882 

50 

199.995 

299.979 

399.947 

499.894 

599.815 

1  0 

199.992 

299.970 

399.924 

499.848 

599.733 

10 

199.990 

299.959 

399.896 

499.793 

599.637 

20 

199.986 

299.946 

399.865 

499.729 

599.526 

30 

199.983 

299.932 

399.829 

499.657 

599.401 

40 

199.979 

299.915 

399.789 

499.577 

599.260 

50 

199.974 

299.898 

399.744 

499.488 

599.105 

2  0 

199.970 

299.878 

399.695 

499.391 

598.934 

10 

199.964 

299.857 

399.643 

499.285 

598.750 

20 

199.959 

299.834 

399.586 

499.171 

598.550 

30 

199.952 

299.810 

399.524 

499.049 

598.336 

40 

199.946 

299.783 

399.459 

498.918 

598.106 

50 

199.939 

299.756 

399.389 

498.778 

597.862 

3  0 

199.931 

299.726 

399.315 

498.630 

597.604 

10 

199.924 

299.695 

399.237 

498.474 

597.331 

20 

199.915 

299.662 

399.154 

498.309 

597.043 

30 

199.907 

299.627 

399.068 

498.136 

596.740 

40 

199.898 

299.591 

398.977 

497.955 

596.423 

50 

199.888 

299.553 

398.882 

497.765 

596.091 

4  0 

199.878 

299.513 

398.782 

497.566 

595.744 

10 

199.868 

299.471 

398.679 

497.360 

595.383 

20 

199.857 

299.428 

398.571 

497.145 

595.007 

30 

199.846 

299.383 

398.459 

496.921 

594.617 

40 

199.834 

299.337 

398.343 

496.689 

594.212 

50 

199.822 

299.289 

398.223 

496.449 

593.792 

5  0 

199.810 

299.239 

398.099 

496.200 

593.358 

10 

199.797 

299.187 

397.970 

495.944 

592.909 

20 

199.783 

299.134 

397.837 

495.678 

592.446 

30 

199.770 

299.079 

397.700 

495.405 

591.968 

40 

199.756 

299.023 

397.559 

495.123 

591.476 

50 

199.741 

298.964 

397.413 

494.832 

590.970 

6  0 

199.726 

298.904 

397.264 

494.534 

590.449 

10 

199.710 

298.843 

397.110 

494.227 

589.913 

20 

199.695 

298.779 

396.952 

493.912 

589.364 

30 

199.678 

298.714 

396.790 

493.588 

588.800 

40 

199.662 

298.648 

396.623 

493.257 

588.221 

50 

199.644 

298.579 

396.453 

492.917 

587.628 

7  0 

199.627 

298.509 

396.278 

492.568 

587.021 

10 

199.609 

298.438 

396.099 

492.212 

586.400 

20 

199.591 

298.364 

395.916 

491.847 

585.765 

30 

199.572 

298.289 

395.729 

491.474 

585.115 

40 

199.553 

298.212 

395.538 

491.093 

584.451 

50 

199.533 

298.134 

395.342 

490.704 

583.773 

TABLE   II.       LONG    CHORDS. 

LONG  CHORDS.    §83. 


175 


Degree 
of  Curve. 

2  Stations. 

3  Stations. 

4  Stations. 

5  Stations. 

6  Stations. 

8  0 

199.513 

298.054 

395.142 

490.306 

583.081 

10 

199.492 

297.972 

394.939 

489.900 

582.375 

20 

199.471 

297.888 

394.731 

489.486 

581.654 

30 

199.450 

297.803 

394.518 

489.064 

580.920 

40 

199.428 

297.716 

394.302 

488.634 

580.172 

50 

199.406 

297.628 

394.082 

488.196 

579.409 

9  0 

199.3a3 

297.538 

393.857 

487.749 

578.633 

10 

199.360 

297.446 

393.629 

487.294 

577.843 

20 

199.337 

297.a52 

393.396 

486.832 

577.039 

30 

199.313 

297.257 

393.159 

486.361 

576.222 

40 

199.289 

297.160 

392.918 

485.882 

575.390 

50 

199.264 

297.062 

392.673 

485.395 

574.545 

10  0 

199.239 

296.962 

392.424 

484.900 

573.686 

10 

199.213 

296.860 

392.171 

484.397 

572.813 

20 

199.187 

296.756 

391.914 

483.886 

571.926 

30 

199.161 

296.651 

391.652 

483.367 

571.027 

40 

199.134 

296.544 

391.387 

482.840 

570.113 

50 

199.107 

296.436 

391.117 

482.305 

569.186 

11  0 

199.079 

296.325 

390.843 

481.762 

568.245 

10 

199.051 

296.214 

390.565 

481.211 

567.291 

20 

199.023 

296.100 

390.284 

480.653 

566.324 

30 

198.994 

295.985 

389.998 

480.086 

565.343 

40 

198.964 

295.868 

389.708 

479.511 

564.349 

50 

198.935 

295.750 

389.414 

478.929 

563.341 

12  0 

198.904 

295.630 

389.116 

478.339 

562.321 

10 

198.874 

295.508 

388.814 

477.740 

561.287 

20 

198.843 

295.384 

388.508 

477.135 

560.240 

30 

198.811 

295.259 

388.197 

476.521 

559.180 

40 

198.779 

295.132 

387.883 

475.899 

558.107 

50 

198.747 

295.004 

387.565 

475.270 

557.020 

13  0 

198.714 

294.874 

387.243 

474.633 

555.921 

176        TABLE  III.   TANGENTS  AND  EXTERNALS 


TABLE  III. 

TANGENTS  AND  EXTERNALS  OF  A  ONE-DEGREE 
CURVE. 

FOR  chords  of  100  feet  the  radius  of  a  one-degree  curve  is 
5729.65  feet.  To  find  its  tangent  for  any  intersection  angle  /, 
we  have  (§  4)  T  —  R  tan.|J,  and  to  find  the  external  (§  85)  b  = 
Ttan.  £  /.  By  these  formulae  this  table  is  computed. 

To  find  T  and  b  for  a  curve  of  any  other  degree  (chords  100 
feet),  divide  the  tabular  values  for  the  proper  intersection  angle 
by  the  number  of  degrees,  whole  or  fractional,  designating  the 
curve.  Thus,  to  find  T  and  b  for  a  3°  20'  curve  we  divide  the 
proper  tabular  values  by  3£.  This  process  supposes  the  radii  of 
curves  to  be  inversely  proportional  to  their  degrees.  ,  This  is  not 
strictly  true,  as  may  be  seen  by  referring  to  Table  I.  Thus  the 
radius  of  a  10°  curve  is  greater  than  one-tenth  the  radius  of  a  1° 
curve.  The  values  of  T  and  b  obtained  as  above  will,  therefore, 
be  too  small,  and  the  corrections  to  be  applied  will  always  be  ad- 
ditive. When  thought  to  be  necessary,  these  corrections  may  be 
obtained  from  Table  IV.  ;  but,  in  the  ordinary  use  of  such  a  table, 
they  may  be  disregarded. 

When  the  intersection  angle  of  a  proposed  curve  is  known,  and 
one  of  the  three  quantities  R,  T,  and  b  is  known  or  assumed,  the 
other  two  may  be  obtained  from  the  table.  Thus,  if  we  have  /  = 
48°  45'  and  the  external  b  =  129  feet,  we  find  from  the  table  for 
this  value  of  /,  b  —  560.7.  Then  we  have  the  degree  of  the  pro- 


posed  curve  =  1°  x  ^~=  =  4C.346  =  4°  20',  nearly.     Also  for  a  1° 
i/oy 

curve  the  table  gives  T  —  2596.1  ;  so  that  for  the  proposed  curve 

2596  1 
T  =         '    =  599.1.    In  a  similar  way,  if  the  tangent  of  a  pro- 

^8" 

posed  curve  is  known  or  assumed,  the  degree  of  the  curve  and  its 
external  can  be  found. 


OF   A   ONE   DEGREE   CURVE. 


177 


I. 

T. 

&. 

I. 

T. 

b. 

I. 

T. 

&. 

!• 

50.0 

.22 

6' 

300.3 

7.86 

11° 

551.7 

26.50 

5> 

54.2 

.26 

5' 

304.5 

8.08 

5' 

555.9 

26.90 

10 

58.3 

.30 

10 

808.6 

8.31 

10 

560.1 

27.31 

15 

62.5 

.34 

15 

312.8 

8.53 

15 

564.3 

27.72 

20 

66.7 

.39 

20 

317.0 

8.76 

20 

568.5 

28.14 

25 

70.8 

.44 

25 

321.2 

8.99 

25 

572.7 

28.55 

30 

75.0 

.49 

30 

325.4 

9.23 

30 

576.9 

28.97 

85 

79.2 

.55 

35 

329.5 

9.47 

35 

581.2 

29.40 

40 

83.3 

.61 

40 

333.7 

9.71 

40 

585.4 

29.82 

45 

87.5 

.67 

45 

337.9 

9.95 

45 

589.6 

30.25 

50 

91.7 

.73 

50 

342.1 

10.20 

50 

593.8 

30.69 

55 

95.8 

.80 

55 

346.3 

10.45 

55 

598.0 

31.12 

2 

100.0 

.87 

7 

350.4 

10.71 

12 

602.2 

31.56 

5 

104.2 

.95 

5 

354.6 

10.98 

5 

606.4 

32.00 

10 

108.3 

1.02 

10 

358.8 

11.22 

10 

610.6 

32.45 

15 

112.5 

1.10 

15 

363.0 

11-49 

15 

614.9 

32.90 

20 

116.7 

1.19 

20 

367.2 

11  -J5 

20 

619.1 

33.35 

25 

120.9 

1.27 

25 

371.4 

12.02 

25 

623.3 

33.80 

30 

125.0 

1.36 

30 

375.5 

12.29 

30 

627.5 

34.26 

35 

129.2 

1.46 

35 

379.7 

12.57 

35 

631.7 

34.72 

40 

ias.4 

1.55 

40 

383.9 

12-85 

40 

635.9 

35.19 

45 

137.5 

1.65 

45 

388.1 

13.13 

45 

640.2 

35.65 

50 

141.7 

1.75 

50 

392.3 

13.41 

50 

644.4 

36.12 

55 

145.9 

1.86 

56 

396.5 

13-70 

55 

648.6 

36.59 

3 

150-0 

1.96 

8 

400.7 

13.99 

13 

652.8 

37.07 

5 

154.2 

2.07 

5 

404.8 

14.28 

5  i    657.0 

37.55 

10 

158.4 

2.19 

10 

409.0 

14.58 

10  !    661.3 

38.03 

15 

162.5 

2.31 

15 

413.2 

14.88 

15       665.5 

38.52 

20 

166.7 

2.42 

20 

417.4 

15.18 

20  S    669.7 

39.01 

25 

170.9 

2.55 

25 

421.6 

15.49 

25       673.9 

39.50 

30 

175.1 

2.67 

30 

425.8       15.80 

30       678.1 

39.99 

35 

179.2 

2.80 

35 

430.0 

16.11 

35       682.4 

40.49 

40 

183.4 

2.93 

40 

434.2 

16.43 

40       686.6 

40.99 

45 

187.6 

3.07 

45 

438.4 

16.74 

45       690.8 

41.50 

60 

191.7 

3.21 

50 

442.5 

17.07 

50       695.1 

42.00 

55 

195.9 

3.35 

55 

446.7 

17.39 

55       699.3 

42.51 

4 

200.1 

3.49 

9 

450.9 

17.72 

14 

703.5 

43.03 

5 

204.3 

3.64 

5 

455.1  i     18.05 

5 

707.7 

43.55 

10 

208.4 

3.79 

10 

459.3       18.38 

10 

712.0 

44.07 

15 

212.6 

3.94 

15 

463.5       18.72 

15 

716.2  1    44.59 

20 

216.8 

4.10 

20 

467.7       19.06 

20 

720.4       45.12 

25 

220.9 

4.26 

25 

471.9       19.40 

25 

724.7       45.65 

30 

225.1 

4.42 

30 

476.1       19.75 

30 

728.9 

46.18 

35 

229.3 

4.59 

35 

480.3 

20.10 

35 

733.1 

46.71 

40 

233.5 

4.75 

40 

484.5 

20.45 

40 

737.4  i    47.25 

45 

237.6 

4.93 

45 

488.7       20.80 

45 

741.6       47.80 

50 

241.8 

5.10 

50 

492.9  !    21.16 

50 

745.8       48.34 

55 

246.0 

5.28 

55 

497.1       21.52 

55 

750.1       48.89 

5 

250.2 

5.46 

10 

501.3       21.89 

15 

754.3       49.44 

5 

254.3 

5.64 

5 

505.5       22.25 

5 

758.6       50.00 

10 

258.5 

5.8S 

10 

509.7       22.62 

10 

762.8       50.55 

15 

262.7 

6.02 

15 

513.9       23.00 

15 

767.0       51.12 

20 

266.9 

6.21 

20 

518.1       23.37 

20 

771.3  i    51.68 

25 

271.0 

6.41 

25 

522.3       23.75 

25 

775.5       52.25 

30 

275.2 

6.61 

30 

526.5       24.14 

30 

779.8  i     52.82 

35 

279.4 

6.81 

35 

530.7       24.52 

35 

784.0       53.39 

40 

283.6 

7.01 

40 

534.9       24.91 

40 

788.3 

53.97 

45 

287.7 

7.22 

45 

539.1       25.30 

45 

792.5 

54.55 

50 

291.9 

7.43 

50 

543.3       25.70 

50 

796.8 

55.13 

55 

296.1 

7.65 

55 

547.5       26.10 

55 

801.0 

56.72 

13 


178 


TABLE  ITT.   TANGENTS  AND  EXTERNALS. 


I. 

T. 

&. 

I. 

T. 

&. 

I. 

T.     ft. 

IB- 

805.2 

56.31 

21° 

1061.9 

97.58 

26° 

1322.8   150.7 

S' 

809.5 

56.90 

5' 

1066.2 

98.36 

5' 

1327.2   151.7 

10 

813.7 

57.50 

10 

1070.6 

99.15 

10 

1331.6  |  152.7 

15 

818.0 

58.10 

15 

1074.9 

99.95 

15 

1336.0   153.7 

20 

822.3 

58.70 

20 

1079.2 

100.7 

20 

1340.4   154.7 

25 

826.5 

59.31 

25 

1083.5 

101.5 

25 

1344.8   155.7 

30 

830.8 

59.91 

30 

1087.8 

102.3 

30 

1349.2   156.7 

35 

835.0 

60.53 

35 

1092.1 

103.2 

35 

1353.6   157.7 

40 

839.3 

61.14 

40 

1096.4 

104.0 

40 

1358.0   158.7 

45 

843.5 

61.76 

45 

1100.8 

104.8 

45 

1362.4   159.7 

50 

847.8 

62.38 

50 

1105.1 

-105.6 

50 

1366.8   160.8 

55 

852.0 

63.01 

55 

1109.4 

106.4 

55 

1371.2 

.161.  8 

17 

856.3 

63.63 

22 

1113.7 

107.2 

27 

1375.6   162.8 

5 

860.6 

64.27 

5 

1118.1 

108.1 

5 

1380.0   163.8 

10 

864.8 

64.90 

10 

1122.4 

108.9 

10 

1384.4  !  164.9 

15 

869.1 

65.54 

15 

1126.7 

109.7 

15 

1388.8  i  165.9 

20 

873.3 

66.18 

20 

1131.0 

110.6 

20 

1393.2   167.0 

25 

877.6 

66.82 

25 

1135.4 

111.4 

25 

1397.6  '  168.0 

30 

881.9 

67.47 

30 

1139.7 

112.3 

30 

1402.0 

169.0 

35 

886.1 

68.12 

35 

1144.0 

113.1 

35 

1406.5 

170.1 

40 

890.4 

68.77 

40 

1148.4 

113.9 

40 

1410.9 

171.2 

45 

894.7 

69.43 

45 

1152.7 

114.8 

45 

1415.3 

172.2 

50 

898.9 

70.09 

50 

1157.0 

115.7 

50 

1419.7 

173.3 

55 

903.2 

70.75 

55 

1161.4 

116.5 

55 

1424.1 

174.3 

18 

907.5 

71.42 

23 

1165.7 

117.4 

28 

1428.6 

175.4 

5 

911.8 

72.09 

5 

1170.1 

118.2 

5 

1433.0 

176.5 

10 

916.0 

72.76 

10 

1174.4 

119.1 

10 

1437.4 

177.6 

15 

920.3 

73.44 

15 

1178.7 

120.0 

15 

1441.8 

178.6 

20 

924.6 

74.12 

20 

1183.1 

120.9 

20 

1446.3 

179.7 

25 

928.9 

74.80 

25 

1187.4 

121.7 

25 

1450.7 

180.8 

30 

933.1 

75.49 

30 

1191.8 

122.6 

30 

1455.1 

181.9 

35 

937.4 

76.18 

35 

1196.1 

123.5 

35 

1459.6 

183.0 

40 

941.7 

76.87 

40 

1200.5 

124.4 

40 

1464.0 

184.1 

45 

946.0 

77.57 

45 

1204.8 

125.3 

45 

1468.5 

185.2 

50 

950.2 

78.26 

50 

1209.2 

126.2 

50 

1472.9 

186.3 

55 

954.5 

78.97 

55 

1213.5 

127.1 

55 

1477.3 

187.4 

19 

958.8 

79.67 

24 

1217.9 

128.0 

29 

1481.8 

188.5 

5 

963.1   80.38 

5  1222.2 

128.9 

5 

1486.2 

189.6 

10 

967.4 

81.09 

10 

1226.6 

129.8 

10 

1490.7 

190.7 

15 

971.7 

81.81 

15 

1230.9 

130.7 

15 

1495.1 

191.9 

20 

976.0   82.53 

20  1235.3 

131.7 

20 

1499.6 

193.0 

25 

980.2  !  83.25 

25 

1239.7 

132.6 

25 

1504.0 

194.1 

30 

984.5   83.97 

30 

1244.0 

133.5 

30 

1508.5 

195.2 

35 

988.8   84.70 

35 

1248.4 

134.4 

35 

1512.9 

196.4 

40 

993.1 

85.43 

40 

1252.8 

135.4 

40 

1517.4 

197.5 

45 

997.4 

86.17 

45 

1257.1 

136.3 

45 

1521.9 

198.7 

50 

1001.7 

86.90 

50 

1261.5 

137.2 

50 

1526.3 

199.8 

55 

1006.0 

87.64 

55 

1265.9 

138.2 

55 

1530.8 

201.0 

20 

1010.3 

88.39 

25 

1270.2 

139.1 

30 

1535.3 

202.1 

5 

1014.6 

89.14 

5 

1274.6 

140.1 

5 

1539.7 

203.3 

10 

1018.9 

89.89 

10 

1279.0 

141.0 

10 

1544.2 

204.4 

15 

1023.2 

90.64 

15 

1283.4 

142.0 

15 

1548.7 

205.6 

20 

1027.5 

91.40 

20 

1287.7 

142.9 

20 

1553.1 

206.8 

25 

1031.8 

92.16 

25 

1292.1 

143.9 

25 

1557.6 

207.9 

30 

1036.1 

92.92 

30 

1296.5 

144.9 

30 

1562.1 

209.1 

35 

1040.4 

93.69 

35 

1300.9 

145.8 

35 

1566.6 

210.3 

40 

1044.7 

94.46 

40 

1305.3 

146.8 

40 

1571.0 

211.5 

45 

1049.0 

95.24 

45 

1309.6 

147.8 

45 

1575.5 

212.7 

50 

1053.3 

96.01 

50 

1314.0 

148.7 

50 

1580.0 

213.9 

55 

1057.6 

96.79 

55 

1318.4 

149.7 

55 

1584.5 

215.1 

OF  A  ONE  DEGREE  CURVE. 


179 


I 

T 

&. 

I. 

T. 

6. 

I. 

T. 

b. 

31° 

1589.0 

216.2 

36°    1861.7 

294.9 

41° 

2142.2 

387.4 

5' 

1593.5 

217.5 

5>  1866.3 

296.3 

5' 

2147.0 

389.0 

10 

1598.0 

218.7 

10  |  1870.9 

297.7 

10 

2151.7 

390.7 

15 

1602.4 

219.9 

15  i  1875.5 

299.1 

15 

2156.5 

392.4 

20 

1606.9 

221.1 

20  1880.1 

300.6 

20 

2161.2 

394.1 

25 

1611.4 

222.3 

25  1884.7 

302.0 

25 

2166.0 

395.7 

30 

1615.9 

223.5 

30  1889.4 

303.5 

30 

2170.8 

397.4 

35 

1620.4 

224.7 

35  1894.0 

304.9 

35 

2175.6 

399.1 

40 

1624.9 

226.0 

40  1898.6 

306.4 

40 

2180.3 

400.8 

45 

1629.4 

227.2 

45  1903.2 

307.8 

45 

2185.1 

402.5 

50 

1633.9 

228.4 

50  1907.9 

309.3 

50 

2189.9 

404.2 

55 

1638.4 

229.7 

55 

1912.5 

310.8 

55 

2194.6 

405.9 

32 

1643.0 

230.9 

37 

1917.1 

312.2 

42 

2199.4 

407.6 

5 

1647.5 

232.1 

5  1921.7 

313.7 

5 

2204.2 

409.4 

10 

1652.0 

233.4 

10  1926.4 

315.2 

10 

2209.0 

411.1 

15 

1656.5 

234.6 

15  1931.0   316.6 

15 

2213.8 

412.8 

20 

1661.0 

235.9 

20  1935.7  I  318.1 

20 

2218-6 

414.5 

25 

1665.5 

237.2 

25  1940.3   319.6 

25 

2223.3 

416  3 

30 

1670.0 

238.4 

30 

1945.0  !  321.1 

30 

2228.1 

418.0 

35 

1674.6 

239.7 

35 

1949.6   322.6 

35 

2232.9 

419.7 

40 

1679.1 

241.0 

40 

1954.3 

324.1 

40 

2237.7 

421.5 

45 

1683.6 

242.2 

45 

1958.9 

325.6 

45 

2242.5 

423.2 

50 

1688.1 

243.5 

50 

1963.6 

327.1 

50 

2247.3 

425.0 

55 

1692.7 

244.8 

55 

1968.2 

328.6 

55 

2252.2 

426.7 

33 

1697.2 

246.1 

38 

1972.9 

330.1 

43 

2257.0 

428.5 

5 

1701.7 

247.4 

5 

1977.5 

331.7 

5 

2261.8 

430.3 

10 

1706.3 

248.7 

10 

1982.2 

333.2 

10 

2266.6 

432.0 

15 

1710.8 

250.0 

15 

1986.9 

334.7 

15 

2271.4 

433.8 

20 

1715.3 

251.3 

20 

1991.5 

336.2 

20 

2276.2 

435.6 

25 

1719.9 

252.6 

25 

1996.2 

337.8 

25 

2281.1 

437.4 

30 

1724.4 

253.9 

30 

2000.9 

339.3 

30 

2285.9 

439.2 

35 

1729.0 

255.2 

35 

2005.6 

340.9 

35 

2290.7 

441.0 

40 

1733.5 

256.5 

40 

2010.2 

342.4 

40 

2295.6 

442.7 

45 

1738.1 

257.8 

45 

2014.9 

344.0 

45 

2300.4 

444.5 

50 

1742.6 

259.1 

50 

2019.6 

345.5 

50 

2305.2 

446.4 

55 

1747.2 

260.5 

55 

2024.3 

347.1 

55 

2310.1 

448.2 

34 

1751.7 

261.8 

39 

2029.0 

348.6 

44 

2314.9 

450.0 

5 

1756.3 

263.1 

5  2033.7 

350.2 

5 

2319.8 

451.8 

10 

1760.8 

264.5 

10 

2038.4 

351.8 

10 

2324.6 

453.6 

15 

1765.4 

265.8 

15 

2043.1 

353.4 

15 

2329.5 

455.4 

20 

1770.0 

267.2 

20 

2047.8 

354.9 

20 

2334.3 

457.3 

25 

1774.5 

268.5 

25  2052.5 

356.5 

25 

2339.2 

459.1 

30 

1779.1 

269.9 

30 

2057.2 

358.1 

30 

2344.1 

460.9 

35 

1783.7 

271.2 

35 

2061.9 

359.7 

35 

2348.9 

462.8 

40 

1788.2 

272.6 

40 

2066.6 

361.3 

40 

2353.8 

464.6 

45 

1792.8 

273.9 

45 

2071  £ 

362.9 

45 

2358.7 

466.5 

50 

1797.4 

275.3 

50 

2076.0 

364.5 

50 

2363.5 

468.4 

55 

1802.0 

276.7 

55 

2080.7 

366.1 

55 

2368.4 

470.2 

35 

1806.6 

278.1 

40 

2085.4 

367.7 

45 

2373.3 

472.1 

5 

1811.1 

279.4 

5 

2090.1 

369.3 

5 

2378.2 

473.9 

10 

1815.7 

280.8 

10 

2094.9 

371.0 

10 

2383.1 

475.8 

15 

1820.3 

282.2 

15 

2099.6 

372.6 

15 

2388.0 

477.7 

20 

1824.9 

283.6 

20 

2104.3 

374.2 

20 

2392.8 

479.6 

25 

1829.5 

285.0 

25 

2109.0 

375.8 

25 

2397.7 

481.5 

30 

1834.1 

286.4 

30 

2113.8   377.5 

30 

2402.6 

483.4 

35 

1838.7 

287.8 

35 

2118.5  i  379.1 

35 

2407.5 

485.3 

40 

1843.3 

289.2 

40 

2123.3   380.8 

40 

2412.4 

487.2 

45 

1847.9 

290.6 

45 

2128.0   382.4 

45  i  2417.4 

489.1 

50 

1852.5 

292.0 

50 

2132.7   384.1 

50  2422.3 

491.0 

55 

1857.1 

293.4 

55  i  2137.5   385.7 

55  1  2427.2 

492.9 

180 


TABLE  III.   TANGENTS  AND  EXTERNALS 


I. 

T. 

&. 

I. 

T. 

&. 

I, 

T. 

ft. 

46' 

2432.1 

494.8 

51° 

2732.9 

618.4 

56° 

3046.5 

759.6 

5' 

2437.0 

496.7 

5' 

2738.0 

620.6 

5' 

3051.9 

762.1 

10 

2441.9 

498.7 

10 

2743.1 

622.8 

10 

3057.2 

764.6 

15 

2446.9 

500.6 

15 

2748.8 

625.0 

15 

3062.6 

767.1 

20 

2451.8 

502.5 

20 

2753.4 

627.2 

20 

3067.9 

769.7 

25 

2456.7 

504.5 

25 

2758.5 

629.5 

25 

3073.3 

772.2 

30 

2461.7 

506.4 

30 

2763.7 

631.7 

30 

3078.7 

774.7 

35 

2466.6 

508.4 

35 

2768.8 

633.9 

35 

3084.0 

777.3 

40 

2471.5 

510.3 

40 

2773.9 

636.2 

40 

3089.4 

779.8 

45 

2476.5 

512.3 

45 

2779.1 

638.4 

45 

3094.8 

782.4 

50 

2481.4 

514.3 

50 

2784.2 

640.7 

50 

3100.2 

784.9 

55 

2486.4 

516.2 

55 

2789.4 

642.9 

55 

3105.6 

787.5 

47 

2491.3 

518.2 

52 

2794.5 

645.2 

57 

3110.9 

790.1 

5 

2496.3 

520.2 

5 

2799.7 

647.4 

5 

3116.3 

792.7 

10 

2501.2 

522.2 

10 

2804.9 

649.7 

10 

3121.7 

795.2 

15 

2506.2 

524.1 

15 

2810.0 

652.0 

15 

3127.2 

797.8 

20 

2511.2 

526.1 

20 

2815.2 

654.3 

20 

3132.6 

800.4 

25 

2516.1 

528.1 

25 

2820.4 

656.5 

25 

3138.0 

803.0 

30 

2521.1 

530.1 

30 

2825.6 

658.8 

30 

3143.4 

805.6 

35 

2526.1 

532.1 

35 

2830.7 

661.1 

35 

3148.8 

808.2 

40 

2531.1 

534.1 

40 

2835.9 

663.4 

40 

3154.2 

810.9 

45 

2536.0 

536.2 

45 

2841.1 

665.7 

45 

3159.7 

813.5 

50 

2541.0 

538.2 

50 

2846.3 

668.0 

50 

3165.1 

816.1 

55 

2546.0 

540.2 

55 

2851.5 

670.3 

55 

3170.6 

818.7 

48 

2551.0 

542.2 

£3 

2856.7 

672.7 

58 

3176.0 

821.4 

5 

2556.0 

544.3 

5 

2861.9 

675.0 

5 

3181.4 

824.0 

10 

2561.0 

546.3 

10 

2867.1 

677.3 

10 

3186.9 

826.7 

15 

2566.0 

548.3 

15 

2872.3 

679.6 

15 

3192.4 

829.3 

20 

2571.0 

550.4 

20 

2877.5 

682.0 

20 

3197.8 

832.0 

25 

2576.0 

552.4 

25 

2882.8 

684.3 

25 

3203.3 

834.6 

30 

2581.0 

554.5 

30 

2888.0 

686.7 

30 

3208.8 

837.3 

35 

2586.0 

556.6 

35 

2893.2 

689.0 

35 

3214.2 

840.0 

40 

2591.1 

558.6 

40 

2898.4 

691.4 

40 

3219.7 

842.7 

45 

2596.1 

560.7 

45 

2903.7 

693.8 

45 

3225.2 

845.4 

50 

2601.1 

562.8 

50 

2908.9 

696.1 

50 

3230.7 

848.1 

55 

2606.1 

564.9 

55 

2914.2 

698.5 

55 

3236.2 

850.8 

49 

2611.2 

566.9 

54 

2919.4 

700.9 

59 

3241.7 

853.5 

5 

2616.2 

569.0 

5 

2924.7 

703.3 

5 

3247.2 

856.2 

10 

2621.2 

571.1 

10 

2929.9 

705.7 

10 

3252.7 

858.9 

15 

2626.3 

673.2 

15 

2935.2 

708.1 

15 

3258.2 

861.6 

20 

2631.3 

575.3 

20 

2940.4 

710.5 

20 

3263.7 

864.3 

25 

2636.3 

577.4 

25 

2945.7 

712.9 

25 

3269.2 

867.1 

30 

2641.4 

579.5 

30 

2951-0 

715.3 

30 

3274.8 

869.8 

35 

2646.5 

581.7 

35 

2956.2 

717.7 

35 

3280.3 

872.6 

40 

2651.5 

583.8 

40 

2961.5 

720.1 

40 

3285.8 

875.3 

45 

2656.6 

585.9 

45 

2966.8 

722.5 

45 

3291.4 

878.1 

50 

2661.6 

588.0 

50  2972.1 

725.0 

50 

3296.9 

880.8 

55 

2666.7 

590.2 

55 

2977.4 

727.4 

55 

3302.5 

883.6 

50 

2671.8 

592.3 

55 

2982.7 

729.9 

60 

3308.0 

886.4 

5 

2676.9 

594.5 

5 

2988.0 

732.3 

5 

3313.6 

889.2 

10 

2681.9 

596.6 

10  2993  3 

734.8 

10 

3319.1 

891.9 

15 

2687.0 

598.8 

15 

2998.6 

737.2 

15 

3324.7 

894.7 

20 

2692.1 

600.9 

20 

3003.9 

739.7 

20 

3330.3 

897.5 

25 

2697.2 

603.1 

25 

3009.2 

742.1 

25 

3335.8 

900.3 

30 

2702.3 

605.3 

30 

3014.5 

744.6 

30 

3341.4 

903.2 

35 

2707.4 

607.4 

35 

3019.8 

747.1 

35 

3347.0 

906.0 

40 

2712.5 

609.6 

40 

3025.2 

749.6 

40 

3352.6 

908.8 

45 

2717.6 

611.8 

45 

3030.5 

752.1 

45 

3358.2 

911.6 

50 

2722.7 

614.0 

50 

3035.8 

754.6 

50 

3363.8 

914.5 

55 

2727.8 

616.2 

55 

3041.2 

757.1 

55 

3369.4 

917.3 

OF  A  ONE  DEGREE  CURVE. 


181 


I. 

T. 

&. 

I. 

T. 

&. 

I. 

T. 

6. 

el- 

3375.0 

920.1 

66* 

3720.9 

1102.2 

71° 

4086.9 

1308.2 

s' 

3380.6 

923.0 

& 

3726.8 

1105.4 

5' 

4093.2 

1311.9 

10 

3386.3 

925.8 

10 

3732.7 

1108.6 

10 

4099.5 

1315.6 

15 

3391.9 

928.7 

15 

3738.7 

1111.9 

15 

4105.8 

1319.2 

20 

3397.5 

931.6 

20 

3744.6 

1115.1 

20 

4112.1 

1322.9 

25 

3403.1 

934.5 

25 

3750.6 

1118.4 

25 

4118.4 

1326.6 

80 

3408.8 

937.3 

30 

3756.5 

1121.7 

30 

4124.8 

1330.3 

35 

3414.4 

940.2 

35 

3762.5 

1124.9 

35 

4131.1 

1334.0 

40 

3420.1 

943.1 

40 

3768.5 

1128.2 

40 

4137.4 

1337.7 

45 

3425.7 

946.0 

45 

3774.4 

1131.5 

45 

4143.8 

1341.4 

50 

3431.4 

948.9 

50 

3780.4 

1134.8 

50 

4150.1 

1345.1 

55 

3437.1 

951.8 

55 

3786.4 

1138.1 

55 

4156.5 

1348.8 

62 

3442.7 

954.8 

67 

3792.4 

1141.4 

72 

4162.8 

1352.6 

5 

3448.4 

957.7 

5 

3798.4 

1144.7 

5 

4169.2 

1356.3 

10 

3454.1 

960.6 

10 

3804  4 

1148.0 

10 

4175.6 

1360.1 

15 

3459.8 

963.5 

15 

3810.4 

1151.3 

15 

4182.0 

1363.8 

20 

3465.4 

966.5 

20 

3816.4 

1154.7 

20 

4188.4 

1367.6 

25 

3471.1 

969.4 

25 

3822.4 

1158.0 

25 

4194.8 

1371.4 

30 

3476.8 

972.4 

30 

3828.4 

1161.3 

30 

4201.2 

1375.2 

35 

3482.5 

975.3 

85 

3834.5 

1164.7 

35 

4207.6 

1379.0 

40 

3488.2 

978.3 

40 

3840.5 

1168.1 

40 

4214.0 

1382.8 

45 

3494.0 

981.3 

45 

1171.4 

45 

4220.4 

1386.6 

50 

3499.7 

984.3 

50 

3852  6 

1174.8 

50 

4226.8 

1390.4 

55 

3505.4 

987.3 

55 

3858.6 

1178.2 

55 

4233.3 

1394.2 

63 

3511.1 

990.2 

68 

3864.7 

1181.6 

73 

4239.7 

1398.0 

5 

3516.9 

993.2 

5 

3870.8 

1185.0 

5 

4246.2 

1401.9 

10 

3522.6 

996.2 

10 

3876.8 

1188.4 

10 

4252.6 

1405.7 

15 

3528.4 

999.3 

15 

3882.9 

1191.8 

15 

4259.1 

1409.6 

20 

3534.1 

1002.3 

20 

3889.0 

1195.2 

20 

4265.6 

1413.5 

25 

3539.9 

1005.3 

25 

3895.1 

1198.6 

25 

4272.0 

1417.3 

30 

3545.6 

1008.3 

30 

3901.2 

1202.0 

30 

4278.5 

1421.2 

35 

3551.4 

1011.4 

35 

3907.3 

1205.5 

35 

4285.0 

1425.1 

40 

3557.2 

1014.4 

40 

3913  4 

1208.9 

40 

4291.5 

1429.0 

45 

3562.9 

1017.4 

45 

3919.5 

1212.4 

45 

4298.0 

1432.9 

50 

3568.7 

1020.5 

50 

3925.6 

1215.8 

50 

4304.5 

1436.8 

55 

3574.5 

1023.6 

55 

3931.7 

1219.3 

55 

4311  1 

1440.7 

64 

3580.3 

1026.6 

69 

3937.9 

1222.7 

74 

4317.6 

1444.6 

5 

3586.1 

1029.7 

5 

3944.0 

1226.2 

5 

4324.1 

1448.6 

10 

3591.9 

1032.8 

10 

3950.2 

1229.7 

10 

4330.7 

1452.5 

15 

3597.7 

1035.9 

15 

3956.3 

1233.2 

15 

4337.2 

1456.5 

20 

3603.5 

1039  0 

20 

::962.5 

1236.7 

20 

4343.8 

1460.4 

25 

3609.3 

1042.1 

25 

3968.6 

1240.2 

25 

4350.4 

1464.4 

30 

3615.1 

1045.2 

80 

3974.8 

1243.7 

30 

4356.9 

1468.4 

35 

3621.0 

1048.3 

35 

3981.0 

1247.2 

35 

4363  5 

1472.4 

40 

3626.8 

1051.4 

40 

3987.2 

'1250.8 

40 

4370.1 

1476.4 

45 

3632.6 

1054.5 

45 

3993.3 

1254.3 

45 

4376.7 

1480.4 

50 

3638.5 

1057.7 

50 

3999.5 

1257.9 

50 

4383.3 

1484.4 

55 

3644.3 

1060.8 

55 

4005.7 

1261.4 

55 

4889.9 

1488.4 

65 

3650.2 

1063.9 

70 

4011.9 

1265.0 

75 

4396.5 

1492.4 

5 

3656.1 

1067.1 

5 

4018.2 

1268.5 

5 

4403.1 

1496.5 

10 

3661.9 

1070.2 

10 

4024.4 

1272  1 

10 

4409.8 

1500.5 

15 

3667.8  1073.4 

15 

4030.6 

1275.7 

15 

4416.4 

1504.5 

20 

3673.7  !  1076.6 

20 

4036.8 

1279.3 

20 

4423.1 

1508.6 

25 

3679.5 

1079.7 

25 

4043.1   1282.9 

25 

4429.7  i  1512.7 

30 

3685.4 

1082.9 

30 

4049.3  1286.5 

30 

4436.4  1  1516.7 

35 

3691.3 

1086.1 

35 

4055.6  1290.1 

35 

4443.0  j  1520.8 

40 

3697.2 

1089.3 

40 

4061.8  1293.6 

40  4449.7  !  1524.9 

45 

3703.1 

1092.5 

45 

4068.1   1297.3 

45 

4456.4  !  1529.0 

50 

3709.0 

1095.7 

50 

4074.4  1300.9 

50 

4463.1  !  1533.1 

55 

3715  0 

1099.0 

55 

4080.6  1304.6 

55 

4469.8  1537.3 

182 


TABLE  III.   TANGENTS  AND  EXTERNALS 


I. 

T. 

b. 

I. 

T. 

b. 

I. 

T. 

»• 

76° 

4476.5 

1541.4 

81° 

4893.6 

1805.3 

86* 

5343.0 

2104.7 

5' 

4483.2 

1545.5 

5' 

4900.8 

1810.0 

5' 

5350.8 

2110.0 

10 

4,189.9 

1549.7 

10 

4908.0 

1814.7 

10 

5358.6 

2115.3 

15 

4496.7 

1553.8 

15 

4915  2 

1819.4 

15 

5366.4 

2120.6 

20 

4503.4 

1558  0 

20 

4922.5 

1824.1 

20 

5374.2 

2126.0 

25 

4510.1 

1562.1 

25 

4929.7 

1828.9 

25 

5382.1 

2131.4 

30 

4516.9 

1566.3 

30 

4937.0 

1833.6 

30 

5389.9 

2136.7 

35 

4523.7 

1570.5 

35 

4944.2 

1838.3 

35 

5397.8 

2142.1 

40 

4530.4 

1574.7 

40 

4951  .5 

1843.1 

40 

5405.6 

2147.5 

45 

4537.2 

1578.9 

45 

4958.8 

1847.9 

45 

5413.5 

2152.9 

50 

4544.0 

1583.1 

50 

4966.1 

1852.6 

50 

5421.4 

2158.4 

55 

4550.8 

1587  3 

55 

4973.4 

1857.4 

55 

5429.3 

2163.8 

77 

4557.6 

1591.6 

82 

4980.7 

1862.2 

87 

5437.2 

2169.2 

5 

4564.4 

1595.8 

5 

4988.0 

1867.0 

5 

5445.2 

2174.7 

10 

4571.2 

1600.1 

10 

4995.4 

1871.8 

10 

5453.1 

2180.2 

15 

4578.0 

1604.3 

15 

5C02.7 

1876.7 

15 

5461.0 

2185.6 

20 

4584.8 

1608.6 

20 

5010.0 

1881.5 

20 

5469.0 

2191.1 

25 

4591.7 

1612.9 

25 

5017.4 

1886.3 

25 

5477.0 

2196.6 

30 

4598.5 

1617.1 

SO 

5024.8 

1891.2 

30 

5484.9 

2202.2 

35 

4605.4 

1621.4 

35 

5032.1 

1896.1 

35 

5492.9 

2207.7 

40 

4612.2 

1625.7 

40 

5039.5 

1900.9 

40 

5500.9 

2213.2 

45 

4619.1 

1630.0 

45 

5046.9 

1905.8 

45 

5509.0 

2218.8 

60 

4626.0 

1634.4 

50 

£054.3 

1910.7 

60 

5517.0 

2224.3 

65 

4632.9 

1638.7 

55 

5061.7 

1915.6 

55 

5525.0 

2229.9 

78 

4639  8 

1643.0 

83 

5069.2 

1920.5 

88 

5533.1 

2235.5 

5 

4646.7 

1647.4 

5 

5076.6 

1925.5 

5 

5541.1 

2241  1 

10 

4653.6 

1651.7 

10 

5084.0 

1930.4 

10 

5549.2 

2246.7 

15 

4660  5 

1656.1 

15 

5091.5 

1985.3 

15 

5557.3 

2252.3 

20 

4667.4 

1660  5 

20 

5099.0 

1940.3 

20 

5565.4 

2258.0 

25 

4674  4 

1664.9 

25 

5106.4 

1945.3 

25 

5573.5 

2263.6 

30 

4681.3 

1669.2 

30 

5113.9 

1950.3 

30 

5581.6 

2269.3 

35 

4688.3 

1673.6 

35 

5121.4 

1955.2 

35 

5589.7 

2275.0 

40 

4695.2 

1678.1 

40 

5128.9 

1960.2 

40 

5597.8 

2280.6 

46 

4702.2 

1682.5 

45 

5136.4 

1965.3 

45 

5606.0 

2286.3 

60 

4709.2 

1686.9 

50 

5143.9 

1970.3 

50 

5614.2 

2292.0 

55 

4716.2 

1691.3 

55 

5151.5 

1975.3 

55 

5622.3 

2297.8 

79 

4723  2 

1695.8 

84 

5159.0 

1980.4 

89 

5630.5 

2303.5 

5 

4730.2 

1700.2 

5 

5166.6 

1985.4 

5 

5638.7 

2309.3 

10 

4737.2 

1704.7 

10 

5174.1 

1990.5 

10 

5646.9 

2315.0 

15 

4744.2 

1709.2 

15 

5181.7 

1995.5 

15 

5655.1 

2320.8 

20 

4751.2 

1713.7 

20 

5189.3 

2000.6 

20 

5663.4 

2326.6 

25 

4758.3 

1718.2 

25 

5196.8 

2005.7 

25 

5671.6 

2332.4 

30 

4765.3 

1722.7 

30 

5204.4 

2010.8 

30 

5679.9 

2338.2 

35 

4772.4 

1727.2 

35 

5212.1 

2016.0 

35 

5688.1 

2844.0 

40 

4779.4 

1731.7 

40 

5219.7 

2021.1 

40 

5696.4 

2349.8 

45 

4786.5 

1736.2 

45 

5227.3 

2026.2 

45 

5704.7 

2355.7 

50 

4793.6 

1740.8 

50 

5234.9 

2031.4 

50 

5713.0 

2361.5 

55 

4800.7 

1745.3 

55 

5242.6 

2036.5 

55 

5721.3 

2367.4 

80 

4807.7 

1749.9 

85 

5250.3 

2041.7 

90 

5729.7 

2373.3 

5 

4814.9 

1754.4 

5 

5257.9 

2046.9 

5 

5738.0 

2379.2 

10 

4822.0 

1759.0 

10 

5265.6 

2052.1 

10 

5746.3 

2385.1 

15 

4829.1 

1763.6 

15 

5273.3 

2057.3 

15 

5754.7 

2391.0 

20 

4836.2 

1768.2 

20 

5281.0 

2062.5 

20 

5763.1 

2397.0 

25 

4843.4 

1772.8 

25 

5288.7 

2067.7 

25 

5771.5 

2402.9 

30 

4850.5 

1777.4 

30 

5296.4 

2073.0 

30 

5779.9 

2408.9 

35 

4857.7 

1782.1 

35 

5304.2 

2078.2 

35 

5788.3 

2414.9 

40 

4864.8 

1786.7 

40 

5311.9 

2083.5 

40 

5796.7 

2420.9 

45 

4872.0 

1791.3 

45 

5319.7 

2088.8 

45 

{•805.1 

2426.9 

50 

4879.2 

1796.0 

50 

5327.4 

2094.1 

50 

5813.6 

2432.9 

55 

4886.4 

1800.7 

55 

5335.2 

2099.4 

55 

5822.1 

2438.9 

TABLE   IV. — TABLE   V. 


183 


TABLE   IV. 
CORRECTIONS  FOR  TABLE  III. 


FOR  TANGENTS  ADD 

FOR  EXTERNALS  ADD 

5° 

10° 

15° 

20° 

25° 

30° 

5° 

10° 

15° 

20° 

25° 

30° 

1. 

Curve. 

Curve. 

Curve. 

Curve. 

Curve. 

Curve. 

f. 

Curve. 

Curve. 

Curve. 

Curve. 

Curve. 

Curve. 

0 

0 

10 

.03 

.00 

.10 

.13 

.16 

.19 

10 

.001 

.003 

.004 

.006 

.007    .008 

20 

.06 

.13 

.19 

.26 

.32 

.39 

20 

.005 

.011 

.017 

.022 

.028 

.034 

30 

.10 

.19 

.29 

.39 

.49 

.60 

30 

.013 

.025 

.038 

.051 

.064 

.078 

40 

.13 

.26 

.40 

.53 

.67 

.80 

40 

.023 

.046 

.070 

.093 

.117 

.141 

50 

.17 

.34 

.51 

.68 

.85 

1.02 

50 

.037 

.075 

.112 

.151 

.189 

.227 

j  60 

.21 

.42 

.63 

.84 

1.05 

1.27 

60 

.054 

.111 

.168 

.225 

.283 

.340 

70 

.25 

.51 

.76 

1.02 

1.28 

1.54 

70 

.077 

.159 

.240 

.321 

.403 

.485 

80 

.30 

.61 

.91 

1.22 

1.53 

1.84 

80 

.110 

.220 

.332 

.445 

.558 

.671 

90 

.35 

.72 

1.09 

1.45 

1.83 

2.20 

90 

.145 

.298 

.451 

.603 

.756 

.910 

TABLE  V. 
TURNOUTS  TANGENT  TO  STRAIGHT  MAIN  TRACK. 

Gauge,  g  =  4.708 ;  throw  of  switch-rail,  d  —  .417.     Ordinates 
;o  E  F  for  all  valnes  of  n,  at  the  centre  1.177,  at  quarter  points 

0.883  (§  68). 


Frog  No., 
§52. 

Frog 
Angle  F, 
§52. 

Switch- 
rail  /, 
§65. 

Chord 
B  F, 
§66. 

Radius, 
§67. 

Degree. 

Curving  30-ft. 
rail,  §29. 

m. 

\m. 

4 

o         / 

14    15 

11.21 

37.96 

150.66 

0         / 

38  46 

.747 

.560 

4£ 

12    41 

12.61 

42.63 

190.67 

30  24 

.590 

.443 

5 

11    25 

14.01 

47.31 

235.40 

24  32 

.478 

.358 

5^ 

10    23 

15.41 

52.00 

284.83 

20  13 

.395 

.296 

6 

9    32 

16.81 

56.69 

338.98 

16  58 

.332 

.249 

*t 

8    48 

18.22 

61.38 

397.83 

14  26 

.283 

.212 

7 

8    10 

19.62 

66.08 

461.38 

12  27 

.244 

.183 

74 

7    38 

21.02 

70.78 

529.65 

10  50 

.212 

.159 

8 

7      9 

22.42 

75.47 

602.62 

9  31 

.187 

.140 

B) 

6    44 

23.82 

80.18 

680.31 

8  26 

.165 

.124 

9 

6    22 

25.22 

84.87 

762.70 

7  31 

.148 

.111 

»* 

6      2 

26.62 

89.58 

849.79 

6  45 

.132 

.099 

10 

5    43 

28.02 

94.28 

941.60 

6    5 

.119 

.090 

104 

5    27 

29.42 

98.98 

1038.11 

5  31 

.108 

.081 

11 

5    12 

30.83 

103.68 

1139.34 

5    2 

.099 

.074 

1H 

4    59 

32.23 

108.39 

1245.27 

4  36 

.090 

.068 

12 

4    46 

33.63 

113.09 

1355.90 

4  14 

.083 

.062 

184 


TABLE   VI. — TABLE   VII. 


TABLE  VI. 
LENGTH  OF  CIRCULAR  ARCS  IN  PARTS  OF  RADIUS. 


1 

.01745  32925  19943 

1 

.00029  08882  08666 

1 

.00000  48481  36811 

2 

.03490  65850  39887 

2 

.00058  17764  17331 

2 

.00000  96962  73622 

3 

.05235  98775  59&30 

3 

.00087  26646  25997 

3 

.00001  45444  10433 

4 

.06981  31700  79773 

4 

.00116  35528  34663 

4 

.00001  93925  47244 

5 

.08726  64625  99716 

5 

.00145  44410  43329 

5 

.00002  42406  84055 

6 

.10471  97551  19660 

6 

.00174  53292  51994 

6 

.00002  90888  20867 

7 

.12217  30476  39603 

7 

.00203  62174  60660 

r 

.00003  39369  57678 

8 

.13962  63401  59546 

8 

.00232  71056  69326 

8 

.00003  87850  94489 

9 

.15707  96326  79490 

9 

.00261  79938  77991 

9 

.00004  36332  31300 

TABLE   VII. 

ELEVATION  OF  THE  OUTER  RAIL  ON  CURVES.    §  152. 


De- 
gree. 

V  = 

15. 

V  = 

20. 

V  = 
25. 

V  = 

30. 

V  = 

35. 

y  — 

40. 

V  = 
45. 

V  = 

50. 

V^ 
60. 

V  = 

70. 

V  = 

80. 

0 

.012 

.022 

.034 

.049 

.067 

.088 

.111 

.137 

.197 

.269 

.351 

2 

.025 

.044 

.068 

.099 

.134 

.175 

.222 

.274 

.395 

.537 

.701 

3 

.037 

.066 

.103 

.148 

.201 

.263 

.333 

.411 

.592 

.805 

1.052 

4 

.049 

.088 

.137 

.197 

.268 

.351 

.444 

.548 

.789 

1.074 

5 

.062 

.110 

.171 

.247 

.336 

.438 

.555 

.685 

.986 

6 

.074 

.131 

.205 

.296 

.403 

.526 

.666 

.822 

7 

.086 

.153 

.240 

.345 

.470 

.613 

.776 

.958 

8 

.099 

.175 

.274 

.394 

.537 

.701 

.887 

1.095 

9 

.111 

.197 

.308 

.443 

.604 

.788 

.998 

10 

.123 

.219 

.342 

.493 

.670 

.876 

12 

.160 

.263 

.410 

.591 

.804 

1.050 

14 

.172 

.306 

.478 

.689 

.938 

16 

.197 

.350 

.546 

.787 

1.071 

TABLE  VIII. — CORRECTION   FOR   THE   EARTH'S   CURVATURE.   185 


TABLE  VIII. 


CORRECTION    FOR    THE    EARTH'S   CURVATURE  AND 
FOR  REFRACTION.    §145. 


Z). 

d. 

D. 

d. 

D. 

d. 

D. 

d. 

300 

.002 

1800 

.066 

3300 

.223 

4800 

.472 

400 

.003 

1900 

.074 

3400 

.237 

4900 

.492 

500 

.  .005 

2000 

.082 

3500 

.251 

5000 

.512 

600 

.007 

2100 

.090 

3600 

.266 

5100 

.533 

700 

.010 

2200 

.099 

3700 

.281 

5200 

.554 

800 

.013 

2300 

.108 

3800 

.296 

Imile 

.571 

900 

.017 

2400 

.118 

3900 

.312 

2 

2.285 

1000 

.020 

2500 

.128 

4000 

.328 

3 

5.142 

1100 

.025 

2600 

.139 

4100 

.345 

4 

9.142 

1200 

.030 

2700 

.149 

4200 

.362 

5 

14.284 

1300 

.035 

2800 

.161 

4300 

.379 

6 

20.568 

1400 

.040 

2900 

.172 

4400 

.397 

7 

27.996 

1500 

.046 

3000 

.184 

4500 

.415 

8 

36.566 

1600 

.052 

3100 

.197 

4600 

.434 

9 

46.279 

1700 

.059 

3200 

.210 

4700 

.453 

10 

57.135 

186        TABLE   IX.      RISE   PER  MILE   OF   VARIOUS   GRADES. 


TABLE  IX. 
EISE  PER  MILE  OP  VARIOUS  GRADES, 


Grade 
per 
station. 

Kise  per 
Mile. 

Grade 
per 
Station. 

Else  per 
Mile. 

Grade 
per 

Station 

Kise  per 

Mile. 

Grade 
per 

Station. 

Eise  per 
Mile. 

.01 

.528 

.41 

21.648 

.81 

42.768 

1.21 

63.888 

.02 

1.056 

.42 

22.176 

.82 

43.296 

1.22 

64.416 

.03 

1.584 

.43 

22.704 

.83 

43.824 

1.23 

64.944 

.04 

2.112 

.44 

23.232 

.84 

44.352 

1.24 

65.472 

.05 

2.640 

.45 

23.760 

.85 

44.880 

1.25 

66.000 

.06 

3.168 

.46 

24.288 

.86 

45.408 

1.26 

66.528 

.07 

3.696 

.47 

24.816 

.87 

45.936 

1.27 

67.056 

.08 

4.224 

.48 

25.344 

.88 

46.464 

1.28 

67.584 

.09 

4.752 

.49 

25.872 

.89 

46.992 

1.29 

68.112 

.10 

5.280 

.50 

26.400 

.90 

47.520 

1.30 

68.640 

.11 

5.808 

.51 

26.928 

.91 

48.048 

1.31 

69.168 

.12 

6.336 

.52 

27.456 

.92 

48.576 

1.32 

69.696 

.13 

6.864 

.53 

27.984 

.93 

49.104 

1.33 

70.224 

.14 

7.392 

.54 

28.512 

.94 

49.632 

.34 

70.752 

.15 

7.920 

.55 

29.040 

.95 

50.160 

.35 

71.280 

.16 

8.448 

.56 

29.568 

.96 

50.688 

.36 

71.808 

.17 

8.976 

.57 

30.096 

.97 

51.216 

.37 

72.336 

.18 

9.504 

.58 

30.624 

.98 

51.744 

.38 

72.864 

.19 

10.032 

.59 

31.152 

.99 

52.272 

.39 

73.392 

.20 

10.E60 

.60 

31.680 

1.00 

52.800 

1.40 

73.920 

.21 

11.088 

.61 

32.208 

1.01 

53.328 

1.41 

74.448 

.22 

11.616 

.62 

32.736 

1.02 

53.856 

1.42 

74.976 

.23 

12.144 

.63 

33.264 

1.03 

54.384 

1.43 

75.504 

.24 

12.672 

.64 

33.792 

1.04 

54.912 

1.44 

76.032 

.25 

13.200 

.65 

34.320 

1.05 

55.440 

1.45 

76.560 

.26 

13.728 

.66 

34.848 

1.06 

55.968 

1.46 

77.088 

.27 

14.256 

.67 

35.376 

1.07 

56.496 

1.47 

77.616 

.28 

14.784 

.68 

35.904 

1.08 

57.024 

1.48 

78.144 

.29 

15.312 

.69 

36.432 

1.09 

57.552 

1.49 

78.672 

.30 

15.840 

.70 

36.960 

1.10 

58.080 

1.50 

79.200 

.31 

16.368 

.71 

37.488 

.11 

58.608 

1.51 

79.728 

.32 

16.896 

.72 

38.016 

.12 

59.136 

1.52 

80.256 

.33 

17.424 

.73 

38.544 

.13 

59.664 

1.53 

80.784 

.34 

17.952 

.74 

39.072 

.14 

60.192 

1.54 

81.312 

.35 

18.480 

.75 

39.600 

.15 

60.720 

1.55 

81.840 

.36 

19.008 

.76 

40.128 

.16 

61.248 

1.56 

82.368 

.37 

19.536 

.77 

40.656 

.17 

61.776 

1.57 

82.896 

.38 

20.064 

.78 

41.184 

1.18 

62.304 

1.58 

83.424 

.39 

20.592 

.79 

41.712 

1.19 

62.832 

1.59 

83.952 

.40 

21.120 

.80 

42.240 

1.20 

63.360 

1.60 

84.480 

TABLE   IX.      RISE   PER   MILE   OF   VARIOUS   GRADES.        187 


Grade 
per 
Station. 

Rise  per 
Mile. 

Grade 
per 
Station. 

Rise  per 
Mile. 

Grade 
per 
Station. 

Rise  per 
Mile. 

Grade 
per 
Station. 

Rise  per 
Mile. 

1.61 

85.008 

1.81 

95.568 

2.10 

110.880 

4.10 

216.480 

1.62 

85.536 

1.82 

96.096 

2.20 

116.160 

4.20 

221.760 

1.63 

86.064 

1.83 

•    96.624 

2.30 

121.440 

4.30 

227.040 

1.64 

86.592 

1.84 

97.152 

2.40 

126.720 

4.40 

232.320 

1.65 

87.120 

1.85 

97.680 

2.50 

132.000 

4.50 

237.600 

1.66 

87.648 

1.86 

98.208 

2.60 

137.280 

4.60 

242.880 

1.67 

88.176 

1.87 

98.736 

2.70 

142.560 

4.70 

248.160 

1.68 

88.704 

1.88 

99.264 

2.80 

147.840 

4.80 

253.440 

1.69 

89.232 

1.89 

99.792 

2.90 

153.120 

4.90 

258.720 

1.70 

89.760 

1.90 

100.320 

3.00 

158.400 

5.00 

264.000 

1.71 

90.288 

1.91 

100.848 

3.10 

163.680 

5.10 

269.280 

1.72 

90.816 

1.92 

101.376 

3.20 

168.960 

5.20 

274.560 

1.73 

91.344 

1.93 

101.904 

3.30 

174.240 

5.30 

279.840 

1.74 

91.872 

1.94 

102.432 

3.40 

179.520 

5.40 

285.120 

1.75 

92.400 

1.95 

102.960 

3.50 

184.800 

5.50 

290.400 

1.76 

92.928 

1.96 

103.488 

3.60 

190.080 

5.60 

295.680 

1.77 

93.456 

1.97 

104.016 

3.70 

195.360 

5.70 

300.960 

1.78 

93.984 

1.98 

104.544 

3.80 

200.640 

5.80 

306.240 

1.79 

94.512 

1.99 

105.072 

3.90 

205.920 

5.90 

311.520 

1.80 

95.040 

2.00 

105.600 

4.00 

211.200 

6.00 

316.800 

188 


TABLE   X.       TRIGONOMETRICAL   AND 


TABLE   X. 


TRIGONOMETRICAL  AND  MISCELLANEOUS 
FORMULA. 

LET  A  (fig.  77)  be  any  acute  angle,  and  let  a  perpendicular  B  C 
be  drawn  from  any  point  in  one  side  to  the  other  side.     Then,  if 


Fig.  77. 


the  sides  of  the  right  triangle  thus  formed  are  denoted  by  letters, 
as  in  the  figure,  we  shall  have  these  six  formulae : 


1.       .       i* 
.    sin.  A  =  —  . 
c 

2.    cos.  A  =  -  . 
c 


3. 


0' 


4.  cosec.  A  =  — . 

a 

c 

5.  sec.     A  =  T . 

0 


6.    cot.     A  =  - . 
a 


Solution  of  Right  Angles  (fig.  77). 


7 

8 
9 

10 
11 

Given. 

Sought. 

Formulae. 

a,  c 

a,  ~b 
A,  a 

A,b 
A,c 

A,B,l 

A,B,c 
B,  6,  c 

B,a,c 
B,a,l> 

a                   a              j 

sin.  A  =  -  ,  cos.  B—-,  ~b  —  y(c  +  a)(c  —  a), 
c                   c 

tan.  A  =  -r  ,        cot.  B  =  j-  ,    c  =  \/a*  +  62. 

7?         QO°           A         7.          n  ^^4-     A         n 

sm.  A 

7?           Qfi°             A           ft        •   7i  fan       \           f> 

cos.  A 
B  =  90°  —  A,     a=  c  sin.  A,     ~b  —  c  cos.  A. 

MISCELLANEOUS    FORMULA. 


189 


Solution  of  Oblique  Triangles  (fig.  78). 


Fig.  78. 


12 
13 
14 

15 

16 

17 

18 

Given. 

Sought. 

Formulae. 

A,£,a 
A,  a,  b 

a,  b,  C 

a,b,c 

A,B,C,a 

A,  b,  c 
a,  b,  c 

b 
B 
A       7? 

,      a  sin.  B 

sin.  A 
-o      b  sin.  A 

a 
tan  l(A      B\      (a  ~  b)  tolL  *  (A  +  B} 

A 

area 

area 
area 

a  +  b 

Tf  *  -  !(„.  +  7,  +  C)   «iuiA-  l/(*~~6)  (S~C) 

be 

J  cos  i  4  -  a  /«(*-«)  t«n  4  ^  -  .  /(*-b)(*-C) 

\       be                    V       S(a-a)    ' 

sin  A  -  2Vs(s  -  a)  (s  -  b)(s  -  c) 

I                                    &c 
a2  sin.  5  sin.  (7 

area  —            . 
2  sin.  J. 

area  =  |Jcsin.  J.. 

s  =  %(a  4-  &  +  c),  area  =  \/s  (5—  a)  (s—b)  (s—c). 

General  Trigonometrical  Formulae. 


19  sin.2  A  +  cos.2  ^4  =  1. 

20  sin.  ( A  ±  B)  =  sin.  A  cos.  B  ±  sin.  B  cos.  Jl. 

21  cos.  ( A  ±  B)  =  cos.  A  cos.  ^  T  sin.  A  sin.  i?. 

22  sin.  2  J.  =  2  sin.  A  cos.  A. 

23  cos.  2  J.  =  cos.2  A  —  sin.2  A  =  1  —  2  sin.2  A  =  2  cos.2  A 

24  sin.2  ^i  =  i  —  i  cos.  2  A. 


-1. 


190  TABLE   X.       TRIGONOMETRICAL   AND 

General  Trigonometrical  JTormulce,  (Continued). 


25 
26 

27 
28 
29 
30 
31 

32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 

cos.2  A  = 
sin.  A  + 
sin.  A  — 
cos.  A  + 
cos.  B  — 
sin.2  A  - 
cos.2  A  — 

•$•  +  1  cos.  2  J.. 
sin.  B  =  2  sin.  |  (J.  +  .#)  cos.  i  (^L  —  B). 
sin.  J5  —  2  cos.  £  (A  +  B)  sin.  i  (A  —  .#). 
cos.  B  =  2  cos.  |  (A  +  7?)  cos.  i  (A  —  .#). 
cos.  J.  =  2  sin.  $(A  +  B)  sin.  |  (A  —  #). 
sin.2  B  =  cos.2  .#  —  cos.2  A  =  sin.  (A  +  B)  sin.  (J.  —  B). 
sin.2  ^  =  cos.  (A  +  ^)  cos.  (^4  -  B). 
sin.  J. 

tan.  A  — 
cot.  A  = 
tan.  (A  ± 
tan.  A  ± 

cot.  A  ± 
sin.  A  + 

cos.  A  ' 
cos.  A 

sin.  J.  " 
^       tan.  Jl  ±  tan.  i? 

1:F8ta!uA*f 

cos.  ^4  cos.  B  ' 

,    „           sin.  (A  ±  B) 
cot  B  —  + 

"""  sin.  A  sin.l?" 
sin.  B      tan.  $(A  +  B) 

sin.  A  — 
sin.  A  + 

sin.  5      tan.  ^  (^4  —  B)  ' 
sin.  -D       ,        \i\  _]_  p\ 

cos.  A  + 
sin.  A  + 

o5Ti   ;-    *t4f^ 

Sin<  ^       rot  4  M        ^ 

cos.  B  — 
sin.  A  — 

cos.  A 

sin.  5              .  ,  A       Dx 

cos.  A  + 
sin.  A  — 

cos.  5 
sin>jB       rot  4M  +  Jft 

cos.  B  — 
tan.  $  A  -- 

cot.  i  .A  : 

cos.  J. 
sin.  ^4 

~  1  +  cos.  ^4  * 
sin.  A 

~~  1  —  cos.  A  * 

MISCELLANEOUS   FORMULAE. 

Miscellaneous  Formulae. 


191 


Sought. 

Given. 

Formulas. 

Area  of 

44 

Circle 

Radius  =  r 

Trr2. 

45 

Ellipse 

Semi-axes  =  a  and  b 

vab. 

40 

Parabola 

Chord  =  c,  height  =  h 

f  ch* 

47 

Regular  Polygon 

(  Side  —  a,  number  of  j 
(  sides  =  n                     j 

.  180° 
^  a2  n  cot.  . 
n 

Surface  of 

48 

Sphere 

Radius  =  r 

47rr2. 

40 

Zone 

Radius  =  r,  height  =  h 

2  IT  r  h. 

50 

Spherical  Poly-  j 
gon                 j 

j  Radius  of  sphere  =  r  j 
j  sum  of  angles  =  S     }• 

o    7,^,    o\i  Qn° 

kj       O  —  ^/fr  —  lOyiOU   . 

180° 

[number  of  sides  =  n  j 

Solidity  of 

51 

Prism  or  Cylin-  ) 
der                  j 

Base  =  b,  height  =  h 

i)fc 

52 

Pyramid  or  Cone 

Base  =  £,  height  =  h 

i&^. 

5:* 

Frustum   ofl 

Pyramid    o  r  j. 
Cone 

(  Bases  =  b    and    &i  ,  ) 
j  height  =  h                  j 

W  +  »,4V^). 

54 

Sphere 

Radius  =  r 

J  IT  ?'3. 

55 

Spherical   Seg-  ) 

(  Radii   of    bases  =  r\ 

T,/   2            2           7,2 

ment               j 

(  and  7*1,  height  =  h     \ 

lir/j,(r  +  n  4»t*  )• 

50 

Prolate  Spheroid 

Semi-transverse  axisl 
of  ellipse  =  a 

^ 

57 

Oblate  Spheroid 

Semi-conjugate   axis  | 
.     of  ellipse  =  b 

|*a% 

58 

Paraboloid 

(  Radius  of  base  =  r,  | 

4"  IT  r^fi. 

]  height  =  h                  \ 

TT  =  3.14159  26535  89793  23846  26433  83280. 
Log.  v  =  0.49714  98726  94133  85435  12682  88291. 


*  The  area  of  a  circular  segment  on  railroad  curves,  where  the  chord  is 
very  long  in  proportion  to  the  height,  may  be  found  with  great  accuracy  by 
the  above  formula. 


192  TABLE   X.       TRIGONOMETRICAL    AND 

Miscellaneous  Formulas,  (Continued). 

United  States  Standard  Gallon  =  231       cub.  in.  =  0.133681  cub.  ft. 

Bushel  =  2150.42       "      =1.244456      " 
British  Imperial  Gallon  =277.27384  "      =0.160459      " 

Length  of  Seconds  Pendulum,  at  sea-level,  at  Equator,  39.0152  in. 
"       "        "  "  "        "  "  N.York,  39.1017  " 

"  London,  39,1393  " 

Weight  of  a  Cubic  Foot  of  Pure  Water,  according  to  Rankine  : 
At  39.4°  Fahrenheit,  62.425  Ibs.  ;  at  62°,  62.355  Ibs. 

Figure  of  the  Earth,  Clarke,  Ency.  Brit.  Art.  Geodesy  : 
Equatorial  radius  =  20  926  202  feet, 
Polar  radius          =  20  854  895    " 

Degrees  in  arc  equal  to  radius          57.29578 
Minutes"    "        "     "      «  3437.74677 

Seconds  "    "        "     "      "        206264.80625 

To  change  common  logarithms  into  hyperbolic  multiply  by 
.434  294  48  ;  the  logarithm  of  which  is  9.637  7843. 


x  =  tan.  x  —  -J-  tan.%  +  £  tan.6#  —  |  tan.7o;  +  &c. 

Let  a  =  length  of  a  flat  circular  arc,  c  =  its  chord,  R  —  radius, 
D  =  deflection  angle  for  100  ft.  chords. 

a3  c3 

Then  approximately  a  —  c  =  _  .  p2  =         2=  £  a  sin.2J9=£  c  sm.2I>. 


TABLES  XI.  XII.   HEIGHTS  BY  ANEROID  BAROMETER.  193 


TABLES  XI.   AND   XII. 

HEIGHTS  BY  ANEROID   BAROMETER. 

THESE  tables  facilitate  the  use  of  the  formula  given  below  for 
obtaining  the  difference  of  height  between  two  stations  by  means 
of  the  aneroid  barometer.  The  formula  and  tables  are  taken  from 
No.  12  of  the  Professional  Papers  of  the  Corps  of  Engineers,  U.  S.  A. 
The  aneroid  barometers  used  are  supposed  to  be  adjusted  to  agree 
with  a  mercurial  barometer  at  a  temperature  of  32°  Fahrenheit, 
at  the  level  of  the  sea,  in  latitude  45°.  Frequent  comparisons 
with  a  mercurial  barometer  are  highly  desirable.  Simultaneous 
observations  of  the  barometers  and  of  the  temperature  of  the  air 
are  to  be  made  at  the  two  stations,  or,  if  only  one  barometer  is 
used,  the  observations  should  differ  in  time  as  little  as  possible- 
In  both  cases,  repeated  observations  should  be  made  when  prac- 
ticable. 
Let  Z  —  the  difference  of  height  of  the  two  stations  in  feet. 

"    h  =  the  reading  in  inches  of  the  barometer  at  the  lower  station. 

"  H=  "  «    «   upper      » 

"   t  and  t'  =  the  temperatures  (Fahr.)  of  the  air  at  the  two  stations. 

Then  Z  =  (log.  h  -  log.  H)  x  60384.3  x  (l  +  *  +  *'  ~  64°)  . 

\  900        / 

Table  XI.  contains  the  products  of  60384.3  and  the  logarithms 
of  any  number  of  inches  from  17  to  31,  except  that,  as  the  charac- 
teristic of  all  these  logarithms  is  one,  this  characteristic  is  omitted 
throughout,  because  the  difference  of  any  two  products  is  not  af- 
fected thereby.  Table  XII.  contains  the  values  of  the  fraction  in 
the  last  parenthesis  of  the  formula  for  all  values  of  t  +  t'  from  30° 
to  189°. 

Example.  Readings  at  lower  station  h  =  29.63  in.,  t  =  68° ;  at 
higher  station,  //=  27.21  in.,  t'  =  61°. 

Table  XI.  gives  for  29.63    28485.2 
"       "       "       "    27.21     26250.8 


difference,          2234.4 
Table  XII.  gives  for  129°        .0722 
• .  Z  =  2234.4  x  1.0722  =  2396  feet. 
14 


194 


TABLE  XI.   FOR  ANEROID  FORMULA. 


M 


bb 
5 
X 


o 

CO 


•saqoni 
a  ui 


ppi-HCOt^        (NOit^t^OS        <?*  OD  CO  CO  OC        C*  OS  OS  -i-l  1O 

fe)  Tji  -^  Tt<  ~xf  rt<      Tji^joibio      10  io  10 10  CD      cococococo 


COOpOCO        trJJSCOCO       rH  OS  OS  <? 


•Ka  GO  T— I  CO  1C  1O        JO'TfOC 


csposp-*      p  os  os  co  os 
" 


^£  t-  T— I  CO  »O  lO        >O  ^t1  Ot  O5 

^^^^^^          ^H^So 


t-rH        IrlOlOOOC^        pGOp-^-i-i 

Sffl      ^S^OiO     ercco^cot- 


1OOSCOO5CO 


QT-HG^IOO     opoopiofN 


»O  O        10  O  O  » 


t^COOSIr-CO       «C500-tH«£5CO        rHC 


JOCDOSOW3 


O  Tt1  Tf  Tt<  "^        rt  "^f  >O  iO  O         S  JO  IO  »O  «D         CD  S 


COOt-JOlO        ttJOSCOOSO        COt-^OCCSCO        COl 


1 


CO  CD  CO  C 


oJcocoiH     THTJJOOOCO 

OlOiOOCO       CDCDCOCOCO 


COCOCCCD;D 


iOCDt^QOOS        C5  r-J  <N  CO  ^JJ        IO  CO  t^  00  O 

a  ' 


TABLE   XI.       FOR   ANEROID    FORMULA.  195 


<s>       O 
<K 

<s  ffl 
tl 

~     X 

X'     <tt> 


PQ 


.  CO  JO  CC  <N  T-I   CO  GO  t-  Oi  •<*< 
^COt-t-t—  t-   t-  t-  t-  00  00 


O  00   OO  Oi  5i  O?  O 


000000000 


!t  SO  t»  t-  C-  t-        J~-t-t-t-OO        OOOOOOOOQO        OO§S§ 


-Pt-t-       t-t-ir-t^QD 


OiOOOCCiO 


ng  a 


•^tOCOO«C(N        i   fi  2  S  98        O-rHi-HC 


pt^copoq 


HCO^        b-O 


196 


TABLE  xi.  FOR  ANEROID  FORMULA. 


8 


•esqoui 


p  10  -«t  oo  CD     p  oo  TH  p  co 
G\«M5t5!5t     5<5*5J?»5« 


lOt^T^lOTH  THOpTHO  TtCOplOp 

ffi  i«  2  i2  » 


TH5^COCOCO        (NOOO'OCO 
f  OS  TH  CO  O         1-C5O(?<TJH 


CO"*ppTtl 

^isi^i 

[^Sa?Soso 


coccGOcoco      cot-Tioir 

{-  CO  OS  TH  (J^  CO  T 


THTHCOCOO 


COr-l 
5T-l(M 


»o  co  c- 1- 1*      co  10  co  o  o 

S^S?S    i2gg§5 


1C  O  TH  TH  CO 

GV  -^  1O  I—  GO 
CO  rf  1O  CO  1.- 


OS  G*  O  <N  GO         CO  CO  <N  »O  CO        CO  CO  lO  TH  CO        OS  O  1O  CO  5 


^COCOOSiO         CO(?»THiO-^        t-iOt-Tt<«O 


Or^COtOCO 


00  CO  OS  Ir;  OS        pOOlOpCO 
OrHCOlOt-         OCOMCOO 


lOOOSCOrH 


S^rSSSi      SE£:Sg22ii 


^  Tt  10  t-  a 


D  CO  TH         t-  rt*  C55  »O  OS 
*  •rf  CO         t-  C:  O  O  CO 

is  o  TH      (?<  co  »o  ^r  t- 


ocotooao 


TH  COC75  1OO 

t-OiTHCO»O 
OTH  CO-^<  )O 


lOCOt-QOOS        Or-tC 


lOtO  t-COOS 


TABLE  XI.   FOR  ANEROID  FORMULA. 


197 


w 


GO 
CO 


PQ 
*3 
EH 


f  TH  Tf  CO  CD  »O        Oi  CO  CO  CO  Oi 

D  10  x  T-H  co  10  CD  t^  06  cd  j> 


-THO^CO   00  00  CO  CO  OS   i-it-OOOr- 


£§££ 


£  ~  e5  e*  G 

^StTiCvJOtG 


ng  ni 


CO  TC  TH  Tf  i-l        lOCOt-t-<M        COO5C*OCO 


1581 


gsss 

r:  •::  /. ; 

^ffic^C 


— i  o  0 


<  (N  <?t        C<  (?<  <?*  C 


rHCOOCOO        QO-rHO^CO        t-  t-  CO  Tf  O        CO  O  CO  CO  t- 


<88c<    ctw'^SxN    c^jic^c 


b-OOiCOCO        COOOTfCDCO        lOCOt-t-C 


t2?,z%  S^^gi  sg^g 

<(?»<?»(?«?<        C^C^G*  <N  <?<        C<Ct(?»C 


198         TABLE  XI.   FOR  ANEROID  FORMULA. 


PQ 


^ 

X 

TH 

00 

I 


pCO^C^OD        lOQOCOCOlO        OOOOG01004 

4%4i%  3§§3§  ssi'iii 


O  1C  1 
<?«<? 


p  O*  Tf  CO  t-        GO  GO  GO  X  t-        CD  »O  CO  r-l  GO 


COL-t^COO         CO  GO  GO  »O  GO        t-CO-^COt 
OC^^co't-       OOaOQOOOtr-       COiOCOi-iO 


•  7*  CO  35  ft       O  )O  »O  rH  r 


CO  GO  Oi 
•  i-l  id  O5  C 


-  o» «  o  co      « t-  oo  ?o  p      pcocot^co 
o^5^€Si      °o»Qdooa6      t-iOCOTHOJ 


COOOlOWCO        lOp-r-JOO<?«        r-<  b-  00  SO  O        fH  C 

4SS2SS?i?     2?S3JiSfr      06  06  en  0606     t^if 


-TH         OCOGOt-rH 


{ 


OOlO  t-lOOS 


•flO>         COOTf  Tf  O 


llslll  iiill  lilii  §^§y 


CO  O  CO  Ct  CO   CO  CO  lf5  T)<  GO   O5  »O  GO  f  00   ^  <N  CO  CO  CO 


O  iH  <7»  CO  •<!}«        iOCOt-GOa»        Oi-KNCO^        lOCOl-GOOt 


TABLE    XI.       FOR    ANEROID   FORMULA. 


199 


^ 

N 

w 

bb 
o 

»— H 

X 

CO 


•  "i 
PQ 


OS  lO  00  i>  CO  lOTfOJrHO  i°.t^lf?1~ie^ 

^3JrHt^CC35  TtCDCOOOC*  1O  00  rH  Tjl  «o 

^  CC  CO  <7<  (?5  i— I  TH  O  O  O5  C5  GO  t- t- CO  1C5 

grTHCSeorHlO  COt-GOGOO5  OT^C^WT? 

^«?«»«?«?  «P«?«?»S  5t-gP? 


S§? 
St^g 


00510t-«0  THT 


I  CO  00  TH       O  <0  0»  CO  U 
!(N<7tC«       C*<?*C*£*C 


GCTHO;OO5        GClOGCGCT 

ciT-J<?ioo\ 


OO5«DO5CO 


:>  C?  C:  K3  I-H        CD  TH  «O  O  rP       OOrHTtt-os 

szm* 


r^Wt^OOiO        Oi  O  t^  O  TH       OOTH^COC 

o«Deoo5ior<      ^odw^-c'^     55  8  »Q  *•  « 


OOSTfCD-*        O5OCO<f*CO        OiOCOCOOO       O5  t^  CM  CO  fH 


CTW7 


T 

C*  C<  (?«? 


CO  CM  CO  000       CO»OCOC004       i^r1CC.T110.       tT^°.<??TH. 


O  — rng^CO        ^^2g 
cJwWC^CM        <NC«7 


«OIOTH^CO     coooocoio      eooooooco     IOCOOSTHO 


200 


TABLE   XI.       FOR    ANEROID    FORMULA. 


S 


x 

^      CO 

txl    ^ 
^    oo 

CO 

K  § 


;  n 


OOG^T^COOS        C*  <?*  O5  T*|  1O 

^inSt^ao      oooTcr"^1" 

OOQCOOOOOO        QCGC'C 


OCOCOG^        O  O5t-U 


:>  CO  «.-  T»<         ODOOCOrHCO        0*CftC*COr-( 


^coT- 

S  O  >— I  G< 


•JOTt!        COi-HOOOO         COOCOJOO5         lOTHCOiHCO 

?xl     §TT2Xf-      ^i$5?^S      ffiSS=£{3?2 


OOOO:»OOO          O5«Or-ieO(? 


^OSOSOSOSCO        t-CO^^O         COlCCNOOlO        THt-0?t-G<{ 
s»COg-^Ogi         OOS^-COIO^         G^— (OOOt-         CD  Tt<  CO  T-H  p 

^&SS82S92    *2o«»s    o^iiig?    iii?il 


QOW3O5OOO       rf  CO  CO  O^  CO       b-  lO  rH  CO  CO 


COOSOOlOOO         O5  « 


^iliSi  iiiii 


TABLE    XII.     FOR    ANEROID    FORMULA. 


TABLE     XII 
FOR  ANEROID  FORMULA. 


t  +  V 

t  +  £'—64 

t  +  t' 

t  +  t'—  64 

t  +  t' 

t  +  t'—  64 

t  +  f 

t  +  £>—  64 

900 

900 

900 

900 

30° 

-0.0378 

70°  +0.0067 

110' 

+  0.0511 

150° 

+0.0956 

31 

.0367 

71 

.0078 

111 

.0522 

151 

.0967 

32 

.0356 

72 

.0089 

112 

.0533 

152 

.0978 

33 

.0344 

73 

.0100 

113 

.0544 

153 

.0989 

34 

.0333 

74 

.0111 

114 

.0556 

154 

.1000 

35 

.0322 

75 

.0122 

115 

.0567 

155 

.1011 

36 

.0311 

76 

.0133 

116 

.0578 

156 

.1022 

37 

.0300 

77 

.0144 

117 

.0589 

157 

.1033 

38 

.0289 

73 

.0156 

118 

.0600 

158 

.1044 

39 

.0278 

79 

.0167 

119 

•0611 

159 

.1056 

40 

.0267 

80 

.0178 

120 

.0622 

160 

.1067 

41 

.0256 

81 

.0189 

121 

.0633 

161 

.1078 

43 

.0244 

82 

.0200 

122 

.0644 

162 

.1089 

43 

.0233 

83 

.0211 

123 

0656 

163 

.1100 

44 

.0222 

84 

.0222 

124 

.0667 

164 

.1111 

45 

.0211 

85 

.0233 

125 

.0678 

165 

.1122 

46 

.0200 

86 

0244 

126 

.0689 

166 

.1133 

47 

.0189 

87 

.0256 

127 

.0700 

167 

.1144 

48 

.0178 

88 

.0267 

128 

.0711 

16S 

.1156 

49 

.0167 

89     .0278 

129 

.0722 

169 

.1167 

50 

.0156 

90     .0289 

130 

.0733 

170 

.1178 

51 

.0144 

91     .0300 

131 

.0744 

171 

.1189 

52 

.0133 

92 

.0311 

132 

.0756 

172 

.1200 

53 

.0122 

93 

.0322 

133 

.0767 

173 

.1211 

51 

.0111 

94 

.0333 

134 

.0778 

174 

.1222 

55 

.0100 

95 

.0344 

135 

.0789 

175 

.1233 

56 

.0089 

96 

.0356 

136 

.0800 

176 

.1244 

57 

.0078 

97 

.0367 

137 

.0811 

177 

.1256 

58 

.0067 

98 

.0378 

138 

.0822 

178 

.1267 

59 

.0056 

99 

.0389 

139 

.0833 

179 

.1278 

60 

.0044 

100 

.0400 

140 

.0844 

180 

.1289 

61 

.0033 

101 

.0411 

141 

.0856 

181 

.1300 

62 

.0022 

102 

.0422 

142 

.0867 

182 

.1311 

63 

-0.0011 

103 

.0433 

143 

.0878 

183 

.1322 

64 

.0030 

104 

.0444 

144 

.0889 

184 

.1333 

65 

+  0.0011 

105 

.0456 

145 

.0900 

185 

.1344 

66 

.0022 

106 

.0467 

146 

.0911 

186 

.1356 

67 

.0033 

107 

.0478 

147 

.0922 

187 

.1367 

68 

.0044 

108 

.0489 

148 

.0933 

188 

,1378 

69 

+  0.0056 

109 

+  0.0500 

149 

+  0.0944 

189 

+0.1389 

TABLE     XIII. 

SQUARES,    CUBES,    SQUARE   ROOTS,    CUBE  ROOTS, 
AND   RECIPROCALS   OF   NUMBERS. 

FROM   1    TO    1054. 


204      TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square  Eoots. 

Cube  Koots. 

Reciprocals. 

1 

1 

1 

1.0000000 

1.0000000 

1.000000000 

2 

4 

8 

1.4142136 

1.2599210 

.500000000 

3 

9 

27 

1.7320508 

1.4422496 

.333333333 

4 

16 

64 

2.0000000 

1.5874011 

.250000000 

5 

25 

125 

2.2360680 

1.7099759 

.200000000 

6 

36 

216 

2.4494897 

1.8171206 

.166666667 

7 

49 

343 

2.6457513 

1.9129312 

.142857143 

8 

64 

512 

2.8284271 

2.0000000 

.125000000 

9 

81 

729 

3.0000000 

2.0800837 

.111111111 

10 

100 

1000 

3.1622777 

2.1544347 

.100000000 

11 

121 

1331 

3.3166248 

2.2239801 

.090909091 

12 

144 

1728 

3.4641016 

2.2894286 

.083333333 

13 

169 

2197 

3.6055513 

2.3513347 

.076923077 

14 

196 

2744 

3.7416574 

2.4101422 

.0714^571 

15 

225 

3375 

3.8729833 

2.4662121 

.060060067 

16 

256 

4096 

4.0000000 

2.5198421 

.002500000 

17 

289 

4913 

4.1231056 

2.5712816 

.058823529 

18 

324 

5832 

4.2426407 

2.6207414 

.055555556 

19 

361 

6859 

4.3588989 

2.6684016 

.052631579 

20 

400 

8000 

4.4721360 

2.7144177 

.050000000 

21 

441 

9261 

4.5825757 

2.7589243 

.047619048 

22 

484 

10648 

4.6904158 

2.8020393 

.045454545 

23 

529 

12167 

4.7958315 

2.8438670 

.043478261 

24 

576 

13824 

4.8989795 

2.8844991 

.041060067 

25 

6£5 

15625 

5.0000000 

2.9240177 

.040000000 

26 

676 

17576 

5.0990195 

2.9624960 

.038461538 

27 

729 

19683 

5.1961524 

3.0000000 

.037037037 

28 

784 

21952 

5.2915026 

3.0365889 

.035714286 

29 

841 

24389 

5.3851648 

3.0723168 

.034482759 

30 

900 

27000 

5.4772256 

3.1072325 

.033333333 

31 

961 

29791 

5.5677644 

3.1413806 

.032258065 

32 

1024 

32768 

5.6568542 

3.1748021 

.031250000 

33 

1089 

35937 

5.7445626 

3.2075343 

.030303030 

34 

1156 

39304 

5.8309519 

3.2396118 

.029411765 

35 

1225 

42875 

5.9160798 

3.2710663 

.028571429 

36 

1296 

46656 

6.0000000 

3.3019272 

.027777778 

37 

1369 

50653 

6.0827625 

3.3322218 

.027027027 

38 

1444 

54872 

6.1644140 

3.3619754 

.026315789 

39 

1521 

59319 

6.2449980 

3.3912114 

.025641026 

40 

1600 

64000 

6.3245553 

3.4199519 

.025000000 

41 

1681 

68921 

6.4031242 

3.4482172 

.024390244 

42 

1764 

74088 

6.4807407 

3.4760266 

.023809524 

43 

1849 

79507 

6.5574385 

3.5033981 

.023255814 

44 

1936 

85184 

6.6332496 

3.5303483 

.022727273 

45 

2025 

91125 

6.7082039 

3.5568933 

.022222222 

46 

2116 

97336 

6.7823300 

3.5830479 

.021739130 

47 

2209 

103823 

6.8556546 

3.60882G1 

.021276600 

48 

2304 

110592 

6.9282032 

3.6342411 

.020833333 

49 

2401 

117649 

7.0000000 

3.6593057 

.020408163 

50 

2500 

125000 

7.0710678 

3.6840314 

.020000000 

51 

2601 

132651 

7.1414284 

3.7084298 

.019607843 

52 

2704 

140608 

7.2111026 

3.7325111 

.019230769 

53 

2809 

148877 

7.2801099 

3.7562858 

.018867925 

54 

2916 

157464 

7.3484692 

3.7797631 

.018518519 

55 

3025 

166375 

7.4161985 

3.8029525 

.018181818 

56 

3136 

175616 

7.4833148 

3.8258624 

.017857143 

57 

3249 

185193 

7.5498344 

3.8485011 

.017543860 

58 

3364 

195112 

7.6157731 

3.8708766 

.017241379 

59 

3481 

205379 

7.6811457 

3.8929965 

.016949153 

60 

3600 

216000 

7.7459667 

3.9148676 

.016666667 

61 

3721 

226981 

7.8102497 

3.930  1972 

.016393443 

6-2 

3844 

238328 

7.8740079 

3.9578915 

.016129032 

CUBE   BOOTS,    AND   RECIPKOCALS. 


205 


No. 

Square*. 

Cubes. 

Square  Roots. 

Cube  Roots 

Reciprocals. 

63 

3969 

250047 

7.9372639 

3.9790571 

.015873016 

64 

4096 

262144 

8.0000000 

4.0000000 

.015625000 

65 

4225 

274625 

8.0622577 

4.0207256 

.015384615 

66 

4356 

287496 

8.1240384 

4.0412401 

.015151515 

67 

4489 

300763 

8.1853528 

4.0615480 

.014925373 

68 

4624 

314432 

8.2462113 

4.0816551 

.014705882 

69 

4761 

328509 

8.3066239 

4.1015661 

.014492754 

70 

4900 

343000 

8.3666003 

4.1212853 

014285714 

71 

5041 

357911 

8.4261493 

4.1408178 

.014084507 

72 

5184 

373248 

8.4852814 

4.1601676 

.013888889 

73 

5329 

389017 

8.5440D37 

4.1793390 

.013698630 

74 

5476 

405224 

8.6023253 

4.1983364 

.013513514 

76 

5625 

421875 

8.6602540 

4.2171633 

.013333333 

76 

6776 

438976 

8.7177979 

4.2358236 

.013157895 

77 

5929 

456533 

8.7749644 

4.2543210 

.012987013 

78 

6084 

474552 

8.8317609 

4.2726586 

.012820513 

79 

6241 

493039 

8.8881944 

4.2908404 

.012658228 

80 

6400 

512000 

8.9442719 

4.3088695 

.012500000 

81 

6561 

531441 

9.0000000 

4.3267487 

.012345679 

82 

6724 

551368 

9.0553851 

4.3444815 

.012195122 

83 

6889 

571787 

9.1104336 

4.3620707 

.012048193 

84 

7056 

592704 

9.1651514 

4.3795191 

.011904762 

85 

7225 

614125 

9.2195445 

4.3968296 

.011764706 

86 

7396 

636056 

9.2738185 

4.4140049 

.011627907 

87 

7569 

658503 

9.3273791 

4.4310476 

.011494253 

88 

7744 

681472 

9.3808315 

4.4479602 

.011363636 

89 

7921 

704969 

9.4339811 

4.4647451 

.011235955 

90 

8100 

729000 

9.4868330 

4.4814047 

.011111111 

91 

8281 

753571 

9.5393920 

4.4979414 

.010989011 

92 

8464 

778688 

9.5916630 

4.5143574 

.010869565 

93 

8649 

804357 

9.6436508 

4.5306549 

.010762688 

94 

8836 

830584 

9.6953597 

4.5468359 

.010638298 

95 

9025 

857375 

9.7467943 

4.5629026 

.010526316 

98 

9216 

884736 

9.7979590 

4.5788570 

.010416667 

97 

9409 

912673 

9.8488578 

4.5947009 

.010309278 

98 

9604 

941192 

9.8994949 

4.6104363 

.010204082 

99 

9801 

970299 

9.9498744 

4.6260650 

.010101010 

100 

10000 

1000000 

10.0000000 

4.6415888 

.010000000 

101 

10201 

1030301 

10.0498756 

4.6570095 

009900990 

102 
103 

10404 
10609 

1061208 
1092727 

10.0995049 
10.1488916 

4.6723287 
4.6875482 

.009803922 
.009708738 

104 

10816 

1124364 

10.1980390 

4.7026694 

.009615385 

105 

11025 

1157625 

10.2469508 

4.7176940 

.009523810 

106 

11236 

1191016 

10.2956301 

4.7326235 

.009433962 

107 
108 

11449 
11664 

1225043 
1259712 

10.3440804 
10.3923048 

4.7474594 
4.7622032 

.009345794 
.009259259 

109 

11881 

1295029 

10.4403065 

4.7768562 

009174312 

110 
111 

12100 
12321 

1331000 
1367631 

10.4880885 
10.5356538 

4.7914199 
4.8058955 

.009090909 
.009009009 

112 
113 
114 
115 
116 
117 
118 

12544 
12769 
12996 
13225 
13456 
13689 
13924 

1404928 
1442897 
1481544 
1520875 
1560896 
1601613 
1643032 

10.5830052 
10.6301458 
10.6770783 
10.7238053 
10.7703296 
10.8166538 
10.8627805 

4.8202845 
4.8345881 
4.8488076 
4.8629442 
4.8769990 
4.8909732 
4.9048681 

.008928571 

.008849558 
.008771930 
.008695652 
.008620690 
.008547009 
.008474576 

119 

14161 

1685159 

10.9087121 

4.9186847 

.008403361 

120 
121 

14400 
14641 

1728000 
1771561 

10.9544512 
11.0000000 

4.9324242 
4.9460874 

.008333333 
.008264463 

122 

14884 

1815848 

11.0453610 

4.9596757 

.008196721 

123 
124 

15129 
15376 

1860S67 
1906624 

11.0905365 
11.1355287 

4.9731898 
4.9866310 

.008130081 
.008064516 

206          TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS, 


Ho. 

Squares, 

Cubes. 

Square  Roots. 

Cube  Roota. 

Reciprocate. 

126 

15625 

1953125 

11.1803399 

5.0000000 

.008000000 

126 

15376 

2000376 

11.2249722 

5.0132979 

.007936508 

127 

16129 

2048383 

11.2694277 

5.0265257 

.007874016 

123 

16384 

2097152 

11.3137085 

5.0396842 

.007812500 

129 

16641 

2146689 

11.3578167 

5.0527743 

.007751938 

130 

16900 

2197000 

11.4017543 

5.0657970 

.007692308 

131 

17161 

2248091 

11.4455231 

5.0787531 

.007633588 

132 

17424 

2299968 

11.4891253 

5.0916434 

.007575758 

133 

17689 

2352637 

11.5325626 

5.1044687 

.007518797 

134 

17956 

2406104 

11.5758369 

5.1172299 

.007462687 

135 

18225 

2460375 

11.6189500 

5.1299278 

007407407 

136 

18496 

2515456 

11.6619038 

5.  1425632 

.007a52941 

137 

18769 

2571353 

11.7046999 

5.1551367 

.007299270 

133 

19044 

2628072 

11.7473401 

5.1676493 

007246377 

139 

19321 

2685619 

11.7898261 

5.1801015 

.007194245 

140 

19600 

2744000 

11.8321596 

6.1924941 

.007142857 

141 

19881 

2803221 

11.8743421 

5.2048279 

007092199 

142 

20164 

2863288 

11.9163753 

5.2171034 

.007042254 

143 

20449 

2924207 

11.9582607 

5.2293215 

.006993007 

144 

20736 

2985984 

12.0000000 

5.2414828 

.006944444 

145 

21025 

3048625 

12.0415946 

5.2535879 

.006896552 

146 

21316 

3112136 

12.0830460 

5.2656374 

.006849316 

147 

21609 

3176523 

12.1243557 

5.2776321 

.006802731 

148 

21904 

3241792 

12.1655251 

5.2895725 

.006756767 

149 

22201 

3307949 

12.2065556 

5.3014592 

.006711409 

160 

22500 

3375000 

12.2474487 

6.3132928 

.006666667 

151 

22801 

3442951 

12.2882057 

5.3250740 

.006622517 

152 

23104 

3511808 

12.3288280 

6.3368033 

.006578947 

153 

23409 

3581577 

12.3693169 

5.3484812 

.006535948 

154 

23716 

3652264 

12.4096736 

5.3601084 

.006493506 

156 

24025 

3723875 

12.4498996 

6.3716854 

.006451613 

156 

24336 

3796416 

12.4899960 

5.3832126 

.006410256 

157 

24649 

3869893 

12.5299641 

6.3946907 

.006369427 

163 

24964 

3944312 

12.5698051 

5.4061202 

.006329114 

159 

25281 

4019679 

12.6095202 

6.4175015 

.006289303 

160 

25600 

4096000 

12.6491106 

6.4288352 

.006250000 

161 

25921 

4173281 

12.6885775 

6.4401218 

.006211180 

162 

26244 

4251528 

12.7279221 

5.4513618 

.006172840 

163 

26569 

4330747 

12.7671453 

5.4625556 

.006134969 

164 

26396 

4410944 

12.8062485 

5.4737037 

.006097,561 

165 

27225 

4492125 

12.8452326 

5.4848066 

.006060606 

166 

27556 

4574296 

12.8840987 

5.4958647 

.006024096 

IK' 

27889 

4657463 

12.9228480 

5.5068784 

.005988024 

163 

28224 

4741632 

12.9614814 

5.5178484 

.005952331 

169 

28561 

4826809 

13.0000000 

5.5287748 

.005917160 

170 

28900 

4913000 

13.0384048 

5.5396583 

.005882353 

171 

29241 

5000211 

13.0766968 

5.5504991 

.005847953 

172 

29584 

5088448 

13.1148770 

5.5612978 

.005813953 

173 

29929 

6177717 

13.1529464 

6.5720546 

.005780347 

174 

30276 

5268024 

13.1909060 

5.5827702 

.005747126 

175 

30625 

5359375 

13.2287566 

6.5934447 

.005714286 

17(5 

30976 

5451776 

13.2664992 

6.6040787 

.005681818 

177 

31329 

5545233 

13.3041347 

5.6146724 

.005649718 

178 

31634 

5639752 

13.3416641 

5.6252263 

.005617978 

179 

32041 

5735339 

13.3790882 

5.6357403 

.005586592 

180 

32400 

5832000 

134164079 

5.6462162 

.005555556 

181 

32761 

5929741 

13.4536240 

5.6566528 

.005524862 

182 

331^ 

6028568 

13.4907376 

66670511 

005494505 

183 

33489 

612&487 

13.5277493 

5.6774114 

005464481 

134 

33856 

6229504 

13.5646600. 

5.6877340 

.005434783 

185 

34225 

6331625 

13.6014705 

5.6980192 

.005405405 

186 

34596 

6434856 

13.6381817 

5.7082675 

.005376344 

CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes 

Square  Roots. 

Cube  Roots. 

Reciprocate. 

187 

34969 

6539203 

13.6747943 

5.7184791 

.005347594 

188 

35344 

6644672 

13.7113092 

6.7286543 

.005319149 

189 

35721 

6751269 

13.7477271 

6.7387936 

005291005 

190 

36100 

6869000 

13.7840488 

6.7488971 

.005263158 

191 

36481 

6967871 

13.8202750 

6.7589652 

-  .005235602 

192 

36S64 

7077888 

13.8564065 

6.7689982 

.005208333 

193 

37249 

7189057 

13.8924440 

6.7789966 

.005181347 

194 

37636 

7301384 

13.9283883 

6.7889604 

.005154639 

195 

38025 

7414875 

13.9642400 

6.7988900 

.005128205 

196 

38416 

7529536 

14.0000000 

6.8087857 

.005102041 

197 

38809 

7645373 

14.0356688 

6.8186479 

.005076142 

198 

39204 

7762392 

14.0712473 

6.8284767 

.005050505 

199 

39601 

7880599 

14.1067360 

6.8382725 

005025126 

200 

40000 

8000000 

14.1421356 

6.8480355 

.005000000 

201 

40401 

8120601 

14.1774469 

6.8577660 

.004975124 

202 

40804 

6242408 

14.2126704 

6.8674643 

.004950496 

203 

41209 

8365427 

14.2478068 

5.8771307 

.004926108 

204 

41616 

8489664 

14.2828569 

6.8867653 

.004901961 

206 

42025 

8615125 

14.3178211 

5.8963685 

.004878049 

206 

42436 

8741816 

14.3527001 

6.9059406   .004854369 

207 

42849 

8869743 

14.3874946 

6.9164817 

004830918 

208 

43264 

8998912 

14.4222051 

6.9249921 

004807692 

209 

43681 

9129329 

14.4568323 

6.9344721 

004784689 

210 

44100 

9261000 

14.4913767 

6.9439220 

.004761906 

211 

44521 

9393931 

14.5258390 

6.9533418 

.004739336 

212 

44944 

9628128 

14.5602198 

6.9627320 

.004716981 

213 

45369 

9663697 

14.6945195 

6.9720926 

.004694836 

214 

45796 

9800344 

14.6287388 

6.9814240 

.004672897 

215 

46225 

9938375 

14.6628783 

5.9907264 

.004651163 

216 

46656 

10077696 

14.6969385 

6.0000000 

.004629630 

217 

47089 

10218313 

14.7309199 

6.0092450 

.004608295 

218 

47524 

10360232 

14.7648231 

6.0184617 

.004587156 

219 

47961 

10503459 

14.7986486 

6.0276502 

.004566210 

220 

48400 

10648000 

14.8323970 

6.0368107 

.004646465 

221 

48841 

10793861 

14.8660687 

6.0459435 

.004524887 

222 

49284 

10941048 

14.8996644 

6.0550489 

.004604606 

223 

49729 

11089567 

14.9331845 

6.0641270 

.004484306 

224 

50176 

11239424 

14.9666295 

6.0731779 

.004464286 

225 

50625 

11390625 

16.0000000 

6.0822020 

.004444444 

226 

61076 

11543176 

15.0332964 

6.0911994 

.004424779 

227 

61529 

11697083 

15.0665192 

6.1001702 

004406286 

228 

51984 

11852352 

15.0996689 

6.1091147 

.004385965 

22S 

52441 

12008989 

15.1327460 

6.1180332 

.004366812 

230 

52900 

12167000 

16.1657509 

6.1269257 

.004347826 

231 

53361 

12326391 

16.1986842 

6.1357924 

.004329004 

232 

63S24 

12487168 

15.2315462 

6.1446337 

.004310345 

233 

64289 

12649337 

15.2643375 

6.1534495 

.004291845 

234 

54756 

12812904 

15.2970586 

6.1622401 

.004273504 

235 

65225 

12977875 

15.3297097 

6.1710058 

.004255319 

236 

65696 

13144256 

15.3622916 

6.1797466 

.004237288 

237 

66169 

13312053 

15.3948043 

6.1884628 

.004219409 

238 

56644 

13481272 

15.4272486 

6.1971544 

.004201681 

239 

67121 

13651919 

15.4596248 

6.2058218 

.004184100 

240 

57600 

13824000 

15.4919334 

6.2144650 

004166667 

241 

58081 

13997521 

15.5241747 

6.2230843 

.004149378 

242 

53564 

14172488 

15.5563492 

6.2316797 

004132231 

243 

69049 

14348907 

15.5884573 

6.2402515 

004115226 

244 

59536 

14526784 

15.6204994 

6.2487998 

004098361 

245 

60025 

14706125 

15.6524758 

6.2573248 

004081633 

246 

60516 

14886936 

15.6843871 

6.2658266 

004065041 

247 

61009 

15069223 

15.7162336 

6.2743054 

.004048583 

248 

61504 

15252992 

15.7480157 

6.2827613 

.004032258 

208          TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

249 

62001 

15438249 

16.7797338 

6.2911946 

.004016064 

250 

62500 

15625000 

15.8113883 

6.2996053 

.004000000 

251 

63001 

15813251 

15.8429795 

6.3079935 

.003984064 

252 

63504 

16003008 

15.8745079 

6.3163596 

.003968254 

253 

64009 

16194277 

15.9059737 

6.3247035 

003952569 

254 

64516 

16387064 

15.9373775 

6.3330256 

.003937008 

255 

65025 

16581375 

15.9687194 

6.3413257 

.003921569 

25,6 

65536 

16777216 

16.0000000 

6.3496042 

.003906250 

257 

66049 

16974593 

16.0312195 

6.  35786  li 

.003891051 

258 

66564 

17173512 

16.0623784 

6.3660968 

.003875969 

259 

67081 

17373979 

16.0934769 

6.3743111 

.003861004 

260 

67600 

17576000 

16.1245155 

6.3825043 

.003846154 

261 

68121 

17779581 

16.1554944 

6.3906765 

.003831418 

262 

68644 

17984728 

16.1864141 

6.3988279 

00381  6794 

263 

69169 

18191447 

16.2172747 

6.4069585 

.003802281 

264 

69696 

183997-44 

16.2480768 

6.4150687 

.003787879 

265 

70225 

18609625 

16.2788206 

6.4231583 

.003773585 

266 

70756 

18821096 

16.3095064 

6.4312276 

.003759398 

267 

71289 

19034163 

16.3401346 

6.4392767 

.003745318 

263 

71824 

19248832 

16.3707055 

6.4473057 

.003731343 

269 

72361 

19465109 

16.4012195 

6.4553148 

.003717472 

270 

72900 

19683000 

16.4316767 

6.4633041 

.003703704 

271 

73441 

19902511 

16.4620776 

6.4712736 

.003690037 

272 

73984 

20123648 

16.4924225 

6.4792236 

.003676471 

273 

74529 

20346417 

16.5227116 

6.4871541 

.003663004 

274 

75076 

20570824 

16.5529454 

6.4950653 

.003649635 

275 

75625 

20796875 

16.5831240 

6.5029572 

.003636364 

276 

76176 

21024576 

16.6132477 

6.5108300 

.003623188 

277 

76729 

21253933 

16.6433170 

6.5186839 

.003610108 

278 

77284 

21484952 

16.6733320 

6.5265189 

.003597122 

279 

77841 

21717639 

16.7032931 

6.5343351 

.003584229 

280 

78400 

21952000 

16.7332005 

6.5421326 

.003571429 

281 

78961 

22188041 

16.7630546 

6.5499116 

.003558719 

282 

79524 

22425768 

16.7928556 

6.5576722 

.003546099 

283 

80089 

22665187 

16.8226038 

6.5654144 

.003533569 

284 

80656 

22906304 

16.8522995 

6.5731385 

.003521  127 

285 

81225 

23149125 

16.8819430 

6.5808443 

.003508772 

286 

81796 

23393656 

16.9115345 

6.5885323 

.003496503 

287 

82369 

23639903 

16.9410743 

6.596*023 

.003484321 

288 

82944 

23887872 

16.9705627 

6.6038545 

.003472222 

289 

83521 

24137569 

17.0000000 

6.6114890 

.003460208 

290 

84100 

24389000 

17.0293864 

6.6191060 

.003448276 

291 

84681 

24642171 

17.0587221 

6.6267054 

.003436426 

292 

85264 

24897038 

17.0880075 

6.6342874 

.003424658 

293 

85849 

25153757 

17.1172428 

6.6413522 

.003412969 

294 

86436 

25412184 

17.1464282 

6.6493998 

003401361 

295 

87025 

25672375 

17.1755640 

6.6569302 

.003389831 

296 

87616 

25934336 

17.2046505 

6.6644437 

.003378378 

297 

88209 

26198073 

17.2330879 

6.67194'  K» 

.003367003 

298 

88804 

26463592 

17.2626765 

6.6794200 

.003355705 

299 

89401 

26730899 

17.2916165 

6.6868831 

.003344482 

300 

90000 

27000000 

17.3205081 

6.6943295 

.003333333 

301 

90601 

27270901 

17.3493516 

6.7017593 

.003322259 

302 

91204 

27543608 

17.3781472 

6.7091729 

.003311258 

303 

91809 

27818127 

17.4068952 

6.7165700 

.003300330 

304 

92416 

28094464 

17,4355958 

6.7239508 

.003289474 

305 

93025 

28372625 

17.4642492 

6.7313155 

.003278689 

306 

93636 

28652616 

17.4928557 

6.7386641 

.003267974 

307 

94249 

28934443 

17.5214155 

6.7459967 

.003257329 

308 

94864 

29218112 

17.5499288. 

6.7533134 

.003246753 

309 

95481 

29503629 

17.5783958 

6.7606143 

.003236246 

310 

96100 

29791000 

17.6068169 

6.7678995 

.003225806 

CUBE   ROOTS,    AND   RECIPROCALS. 


209 


No. 

Sqoaice. 

Cubes. 

Square  Root*  . 

Cube  Roots. 

Reciprocals. 

311 

96721 

30080231 

17.6351921 

6.7751690 

.003215434 

312 

97344 

30371328 

17.6635217 

6.7824229 

.003205128 

313 

97969 

30664297 

17.6918060 

6.7S96613 

.003194888 

314 

98596 

30959144 

17.7200451 

6.7968844 

.003184713 

315 

99225 

31255875 

17.7482393 

6.8040921 

.003174603 

316 

99356 

31554496 

17.7763888 

6.8112847 

.003164557 

317 

100489 

31855013 

17.8044933 

6.8184620 

.003154574 

318 

101124 

32157432 

17.8325545 

6.8256242 

.003144654 

319 

101761 

32461769 

17.86057!! 

6.8327714 

.003134796 

'  320 

102400 

32768000 

17.8885438 

6.8399037 

.003125000 

321 

103041 

33076161 

17.9164729 

6.8470213 

.003115265 

322 

103684 

33386248 

17.9443584 

6.8541240 

.003105590 

323 

104329 

33693267 

17.9722008 

6.8612120 

.003095975 

324 

104976 

34012224 

18.0000000 

6.8682855 

.003086420 

325 

105625 

34328125 

18.0277564 

6.8753443 

.003076923 

326 

106276 

34645976 

18.0554701 

6.8823888 

.003067485 

327 

106929 

34965783 

18.0831413 

6.8894188 

.003058104 

323 

107584 

35287552 

18.1107703 

6.8964345 

.003048780 

329 

108241 

35611289 

18.1383571 

6.9034359 

.003039514 

330 

108900 

35937000 

18.1659021 

$.9104232 

.003030303 

331 

109561 

36264691 

18.1934054 

6.9173964 

.003021148 

332 

110224 

36594368 

18.2208672 

6.9243556 

.003012048 

333 

110889 

36926037 

18.2482876 

6.9313008 

.003003003 

334 

111556 

37259704 

18.2756669 

6.9382321 

.002994012 

335 

112225 

37595375 

18.3030052 

6.9451496 

.002985076 

336 

112896 

37933056 

18.3303028 

6.9520533 

.002976190 

337 

113569 

38272753 

18.3575598 

6.9589434 

.002967359 

333 

114244 

38614472 

18.3847763 

6.9658198 

.002958580 

339 

114921 

38958219 

18.4119526 

6.9726826 

.002949853 

340 

115600 

39304000 

18.4390889 

6.9795321 

.002941176 

341 

116281 

39651821 

18.4661853 

6.9863681 

.002932551 

342 

116964 

40001688 

18.4932420 

6.9931906 

.002923977 

343 

117649 

40353607 

18.5202592 

7.0000000 

.002915452 

344 

118336 

40707584 

18.5472370 

7.0067962 

.002906977 

345 

119025 

41063625 

18.5741756 

7.0135791 

.002898551 

346 

119716 

41421736 

18.6010752 

7.0203490 

.002890173 

347 

120409 

41781923 

18.6279360 

7.0271058 

.002881844 

343 

121104 

42144192 

18.6547581 

7.0338497 

.002873563 

349 

121801 

42508549 

18.6815417 

7.0405806 

.002865330 

350 

122500 

42875000 

18.7082869 

7.0472987 

.002857143 

351 

123201 

43243551 

18.7349940 

7.0540041 

.002849003 

352 

123904 

43614208 

18.7616630 

7.0606967 

.002840909 

353 

124609 

43986977 

18.7882942 

7.0673767 

.002832861 

354 

125316 

44361864 

18.8148877 

7.0740440 

.002824859 

355 

126025 

44738875 

18.8414437 

7.0806988 

.002816901 

356 

126736 

45118016 

18.8679623 

7.0873411 

.002808989 

357 

127449 

45499293 

18.8944436 

7.0939709 

.002801120 

358 

128164 

45882712 

18.9208879 

7.1005885 

.002793296 

359 

128881 

46268279 

18.9472953 

7.1071937 

.002785515 

360 

129600 

46656000 

18.9736660 

7.1137866 

.002777778 

361 

130321 

47045881 

19.0000000 

7.1203674 

.002770083 

362 

131044 

47437928 

19.0262976 

7.1269360 

.002762431 

363 

131769 

47832147 

19.0525589 

7.1334925 

.002754821 

364 

132496 

48228544 

19.0787840 

7.1400370 

.002747253 

365 

133225 

48627125 

19.1049732 

7.1465695 

.002739726 

366 

133956 

49027896 

19.1311265 

7.1530901 

.002732240 

367 

134689 

49430S63 

19.1572441 

7.1595988 

.002724796 

368 

135424 

49836032 

19.1833261 

7.1660957 

.002717391 

369 

136161 

50243409 

19.2093727 

7.1725809 

.002710027 

370 

136900 

50653000 

19.2353841 

7.1790544 

.002702703 

371 

137641 

51064811 

19.2613603 

7.1855162 

.002695418 

372 

133384 

51478848 

19.2873015 

7.1919663 

.002688172 

15 


210         TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

373 

139129 

61895117 

19.3132079 

7.1984050 

.002680986 

374 

139876 

52313624 

19.3390796 

7.2048322 

.002673797 

375 

140625 

52734375 

19.3649167 

7.2112479 

.002666667 

376 

141376 

53157376 

19.3907194 

7.2176522 

.002659574 

377 

142129 

53582633 

19.4164878 

7.2240450 

.002652520 

378 

142884 

54010152 

19.4422221 

7.2304268 

.002645503 

379 

143641 

54439939 

19.4679223 

7.2367972 

.002638522 

380 

144400 

54872000 

19.4935887 

7.2431565 

.002631570 

381 

145161 

55306341 

19.5192213 

7.2495045 

.002624672 

332 

145924 

55742968 

19.5448203 

7.2558415 

.002617801  ' 

383 

146689 

56181887 

19.5703858 

7.2621675 

.002610966 

384 

147456 

56623104 

19.5959179 

7.2034824 

.002604167 

385 

148225 

57066625 

19.6214169 

7.2747864 

.002597403 

386 

148996 

57512456 

19.6468827 

7.2810794 

.002590674 

387 

149769 

57960603 

19.6723156 

7.2873617 

.002583979 

338 

150544 

58411072 

19.6977156 

7.2936330 

.002577320 

389 

151321 

58863869 

19.7230829 

7.2998936 

.002570694 

390 

152100 

59319000 

19.7484177 

7.3061436 

.002564103 

391 

152881 

69776471 

19.7/37199 

7.3123828 

.002557545 

392 

153664 

60236288 

19.79S9899 

7.3186114 

.002551020 

393 

154449 

60698457 

19.8242276 

7.3248295 

.002544529 

394 

155236 

61162934 

19.8494332 

7.3310369 

.002538071 

396 

156025 

61629875 

19.8746069 

7.3372339 

.002531646 

396 

156816 

62099136 

19.8997487 

7.3434205 

.002525253 

397 

157609 

62570773 

19.9248588 

7.3495966 

.002518892 

398 

153404 

63044792 

19.9499373 

7.3557624 

.002512563 

399 

159201 

63521199 

19.9749844 

7.3619178 

.002506266 

400 

160000 

64000000 

20.0000000 

7.3680630 

.002500000 

401 

160801 

64481201 

20.0249844 

7.3741979 

.002493766 

403 

161604 

64964808 

20.0499377 

7.3803227 

.002487562 

403 

162409 

65450827 

20.0748699 

7.3864373 

.002481390 

404 

163216 

65939264 

20.0997512 

7.3925418 

.002475248 

405 

164025 

66430125 

20.1246118 

7.3986363 

.002469136 

406 

164836 

66923416 

20.1494417 

7.4047206 

.002463054 

407 

165649 

67419143 

20.1742410 

7.4107950 

.002457002 

408 

166464 

67917312 

20.1990099 

7.4168595 

.002450980 

409 

167281 

68417929 

20.2237484 

7.4229142 

.002444988 

410 

168100 

68921000 

20.2484567 

7.4289589 

.002439024 

411 

168921 

6&426531 

20.2731349 

7.4349938 

.002433090 

412 

169744 

69934528 

20.2977831 

7.4410189 

.002427184 

413 

170569 

70444997 

20.3224014 

7.4470342 

.002421308 

414 

171396 

70957944 

20.3469899 

7.4530399 

.002415459 

415 

172225 

71473375 

20.3715488 

7.4590359 

.002409639 

416 

173056 

71991296 

20.3960781 

7.4650223 

.002403846 

417 

173889 

72511713 

20.4205779 

7.4709991 

.002398082 

418 

174724 

73034632 

20.4450483 

7.4769664 

002392344 

419 

175561 

73560059 

20.4694895 

7.4829242 

.002386635  ' 

420 

176400 

74088000 

20.4939015 

7.4888724 

,002380952 

421 

177241 

74618461 

20.5182845 

7.4948113 

.002375297 

422 

178084 

75151448 

20.5426386 

7.5007406 

.002369668 

423 

178929 

75686967 

20.5669638 

7.5066607 

.002364066 

424 

179776 

76225024 

20.5912603 

7.5125715 

.002358491 

425 

180625 

76765625 

20.6155281 

7.5184730 

.002352941 

426 

181476 

77308776 

20.6397674 

7.5243652 

.002347418 

427 

182329 

77854483 

20.6639783 

7.5302482 

.002341920 

428 

183184 

78402752 

20.6881609 

7.5361221 

.002336449 

429 

184041 

78953589 

20.7123152 

7.5419867 

.002331002 

430 

184900 

79507000 

20.7364414 

7.5478423 

.002325581 

431 

185761 

80062991 

20.7605395 

7.5536888 

.002320186 

432 

186624 

80621568 

20.7846097 

7.5595263 

.002314815 

433 

187489 

81182737 

20.8086520 

7.5653548 

.002309469 

434 

188356 

81746504 

20.8326667 

7.5711743 

.002304147 

CUBE   HOOTS,    AND   KECIPROCALS. 


No. 

Squares. 

Cubes 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

435 

189225 

82312875 

20.8566536 

7.5769849 

.002298851 

436 

190096 

82881856 

20.8806130 

7.5827865 

.002293578 

437 

190969 

83453453 

20.9045450 

7.5885793 

.002288330 

438 

191844 

84027672 

20.9284495 

7.5943633 

.002283105 

439 

192721 

84604519 

20.9523268 

7.6001385 

.002277904 

440 

193600 

85184000 

20.9761770 

7.6059049 

.002272727 

441 

194481 

85766121 

21.0000000 

7.6116626 

.002267574 

442 

195364 

86350888 

21.0237960 

7.6174116 

.002262443 

443 

196249 

86938307 

21.0475652 

7.6231519 

.002257336 

444 

197136 

87528384 

21.0713075 

7.6288837 

.002252252 

445 

198025 

88121125 

21.0950231 

7.6346067 

.002247191 

446 

198916 

88716536 

21.1187121 

7.6403213 

.002242152 

447 

199809 

89314623 

21.1423745 

7.6460272 

.002237136 

448 

200704 

89915392 

21.1660105 

7.6517247 

.002232143 

449 

201601 

90518849 

21.1896201 

7.6574138 

.002227171 

450 

202500 

91125000 

21.2132034 

7.6630943 

.002222222 

451 

203401 

91733851 

21.2367606 

7.6687665 

.002217295 

452 

204304 

92345408 

21.2602916 

7.6744303 

.002212389 

453 

205209 

92959677 

21.2837967 

7.6800857 

.002207506 

454 

206116 

93576664 

21.3072758 

7.6857328 

.002202643 

455 

207025 

94196375 

21.3307290 

7.6913717 

.002197802 

456 

207936 

94818816 

21.3541565 

7.6970023 

.002192982 

457 

208849 

95443993 

21.3775583 

7.7026246 

.002188184 

458 

209764 

96071912 

21.4009346 

7.7082388 

.002183406 

459 

210681 

96702579 

21.4242853 

7.7138448 

.002178649 

460 

211600 

97336000 

21.4476106 

7.7194426 

.002173913 

461 

212521 

97972181 

21.4709106 

7.7250325 

.002169197 

462 

213444 

98611128 

21.4941853 

7.7306141 

.002164502 

463 

214369 

99252847 

21.5174348 

7.7361877 

.002159827 

464 

215296 

99897344 

21.5406592 

7.7417532 

.002165172 

465 

216225 

100544625 

21.5638587 

7.7473109 

.002150538 

466 

217156 

101194696 

21.5870331 

7.7528606 

.002145923 

467 

218089 

101847563 

21.6101828 

7.7584023 

.002141328 

468 

219024 

102503232 

21.6333077 

7.7639361 

.002136752 

469 

219961 

103161709 

21.6564078 

7.7694620 

.002132196 

470 

220900 

103823000 

21.6794834 

7.7749801 

.002127660 

471 

221841 

104487111 

21.7025344 

7.7804904 

.002123142 

472 

222784 

105154048 

21.7255610 

7.7859928 

.002118644 

473 

223729 

105823817 

21.7485632 

7.7914875 

002114165 

474 

224676 

106496424 

21.7715411 

7.7969745 

002109705 

475 

225625 

107171875 

21.7944947 

7.8024538 

.002105263 

476 

226576 

107850176 

21.8174242 

7.8079254 

.002100840 

477 

227529 

108531333 

21.8403297 

7.8133892 

.002096436 

478 

228484 

109215352 

21.8632111 

7.8188456 

.002092050 

479 

229441 

109902239 

21.8860686 

7.8242942 

.002087683 

480 

230400 

110592000 

21.9089023 

7.8297353 

.002083333 

481 

231361 

111284641 

21.9317122 

7.8351688 

.002079002 

482 

232324 

111980168 

21.9544984 

7.8405949 

.002074689 

483 

233289 

112678587 

21.9772610 

7.8460134 

.002070393 

484 

234256 

113379904 

22.0000000 

7.8514244 

.002066116 

485 

235225 

114084125 

22.0227155 

7.8568281 

.002061856 

486 

236196 

114791256 

22.0454077 

7.8622242 

.002057613 

487 

237169 

115501303 

22.0680765 

7.8676130 

.002053388 

488 

238144 

116214272 

22.0907220 

7.8729944 

.002049180 

489 

239121 

116930169 

22.1133444 

7.8783684 

.002044990 

490 

240100 

117649000 

22.1359436 

7.8837352 

.002040816 

491 

241081 

118370771 

22.1585198 

7.8890946 

.002036660 

492 

242064 

1190954&5 

22.1810730 

7.8944468 

.002032520 

493 

243049 

119823157 

22.2036033 

7.8997917 

.002028398 

494 

244036 

120553784 

22.2261108 

7.9051294 

.002024291 

495 

245025 

121287375 

22.2485955 

7.9104599 

.002020202 

496 

246016 

122023936 

22.2710575 

7.9157832 

.002016129 

TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Boots. 

Reciprocate. 

497 

247009 

122763473 

22.2934968 

7.9210994 

.002012072 

498 

248004 

123505992 

22.3159136 

7.9264085 

.002008032 

499 

249001 

124251499 

22.3383079 

7.9317104 

,002004008 

500 

250000 

125000000 

22.3606798 

7.9370053 

.002000000 

601 

251001 

125751501 

22.3830293 

7.9422931 

.001996008 

602 

252004 

126506008 

22.4053565 

7.9475739 

.001992032 

503 

253009 

127263527 

22.4276615 

7.9528477 

.001988072 

604 

254016 

128024064 

22.4499443 

7.9581144 

.001984127 

505 

255025 

128787625 

22.4722051 

7.9633743 

.001980198 

606 

256036 

129554216 

22.4944438 

7.9686271 

.001976285 

607 

257049 

130323843 

22.5166605 

7.9738731 

.001972387 

603 

258064 

131096512 

22.5388553 

7.9791122 

.001968504 

609 

259081 

131872229 

22.5610283 

7.9843444 

.001964637 

510 

260100 

132651000 

22.5831796 

7.9895697 

.001960784 

611 

261121 

133432S31 

22.6053091 

7.9947883 

.001956947 

612 

262144 

134217728 

22.6274170 

8.0000000 

.001953125 

513 

263169 

135005697 

22.6495033 

8.0052049 

.001949318 

614 

264196 

135796744 

22.6715681 

8.0104032 

.001945525 

615 

265225 

136590875 

22.6936114 

8.0155946 

.001941748 

616 

266256 

137388096 

22.7156334 

8.0207794 

.001937984 

617 

267289 

138188413 

22.7376340 

8.0259574 

.001934236 

518 

268324 

138991832 

22.7596134* 

8.0311287 

.001930502 

619 

269361 

139798359 

22.7815715 

8.0362935 

.001926782 

520 

270400 

140608000 

22.8035085 

8.0414515 

.001923077 

621 

271441 

141420761 

22.8254244 

8.0466030 

.001919386 

622 

272484 

142236648 

22.8473193 

8.0517479 

.001915709 

623 

273529 

143055667 

22.8691933 

8.0568862 

.001912046 

524 

274576 

143877824 

22.8910463 

8.0620180 

.001908397 

625 

275625 

144703125 

22.9128785 

8.0671432 

.001904762 

526 

276676 

145531576 

22.9346899 

8.0722620 

.001901141 

627 

277729 

146363183 

22.9564806 

8.0773743 

.001897533 

528 

278784 

147197952 

22.9782506 

8.0824800 

.001893939 

629 

279841 

148035889 

23.0000000 

8.0875794 

.001890359 

630 

280900 

148877000 

23.0217289 

8.0926723 

.001886792 

631 

281961 

149721291 

23.0434372 

8.0977589 

.001883239 

532 

283024 

150568768 

23.0651252 

8.1028390 

.001879699 

533 

284089 

151419437 

23.0867928 

8.1079128 

.001876173 

534 

285156 

152273304 

23.1084400 

8.1129803 

.001872659 

635 

286225 

153130375 

23.1300670 

8.1180414 

.001869159 

536 

287296 

153990656 

23.1516738 

8.1230962 

.001865672 

637 

288369 

154854153 

23.1732605 

8.1281447 

.001862197 

538 

289444 

155720872 

23.1948270 

8.1331870 

.001858736 

539 

290521 

156590819 

23.2163735 

8.1382230 

.001855288 

640 

291600 

157464000 

23.2379001 

8.1432529 

.001851852 

641 

292681 

158340421 

23.2594067 

8.1482765 

.001848429 

542 

293764 

159220088 

23.2808935 

8.1532939 

.001845018 

543 

294849 

160103007 

23.3023604 

8.1583051 

.001841621 

544 

295936 

160989184 

23.3238076 

8.1633102 

.001838235 

645 

297025 

161878625 

23.3452351 

8.1683092 

.001834862 

646 

298116 

162771336 

23.3666429 

8.1733020 

.001831502 

647 

299209 

163667323 

23.3880311 

8.1782888 

.001828154 

548 

300304 

164566592 

23.4093998 

8.1832695 

.001824818 

549 

301401 

'  165469149 

23.4307490 

8.1882441 

.001821494 

550 

302500 

166375000 

23.4520788 

8.1932127 

.001818182 

551 

303601 

167284151 

23.4733892 

8.1981753 

.001814882 

652 

304704 

168196608 

23.4946802 

8.2031319 

.001811594 

553 

305809 

1691  12377 

23.5159520 

8.2080825 

.001808318 

554 

306916 

170031464 

23.5372046 

8.2130271 

.001805054 

555 

308025 

170953875 

23.5584380 

8.2179657 

.001801802 

656 

309136 

171879616 

23.5796522 

8.2228985 

.001798561 

557 

310249 

172808693 

23.6008474 

8.2278254 

.001795332 

553 

311864 

173741112 

23.6220236 

8.2327463 

.001792115 

CUBE  HOOTS,  AND  KECIPROCALS. 


213 


Ho. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocate. 

569 

312481 

!  74676879 

23.6431808 

8.2376614 

.001788909 

660 

313600 

175616000 

23.6643191 

8.2425706 

.001785714 

661 

314721 

176558481 

23.6854336 

8.2474740 

.001782531 

662 

315844 

177504323 

23.7065392 

8.2523715 

.001779369 

•663 

316969 

178453547 

23.7276210 

8.2572633 

.001776199 

664 

318096 

179406144 

23.7486842 

8.2621492 

.001773050 

665 

319225 

180362125 

23.7697286 

8.2670294 

.001769912 

666 

320356 

181321496 

23.7907545 

8.2719039 

.001766784 

567 

321489 

182284263 

23.8117618 

8.2767726 

.001763668 

568 

322624 

183250432 

23.8327506 

8.2816355 

.001760563 

669 

323761 

184220009 

23.8537209 

8.2864928 

.001767469 

670 

324900 

185193000 

23.8746728 

8.2913444 

.001754386 

671 

326041 

186169411 

23.8956063 

8.2961903 

.001751313 

572 

327184 

187149248 

23.9165215 

8.3010304 

.001748252 

673 

328329 

188132517 

23.9374184 

8.3058651 

.001745201 

674 

329476 

189119224 

23.9582971 

8.3106941 

.001742160 

575 

330625 

190109375 

23.9791576 

8.3155175 

.001739130 

676 

331776 

191102976 

24.0000000 

8.3203353 

.001736111 

677 

332929 

192100033 

24.0208243 

8.3251475 

.001733102 

578 

334084 

193100552 

24.0416306 

8.3299542 

.001730104 

679 

335241 

194104539 

24.0624188 

8.3347553 

.001727116 

680 

336400 

195112000 

24.0831891 

8.3395509 

.001724138 

681 

337561 

196122941 

24.1039416 

8.3443410 

.001721170 

682 

338724 

197137368 

24.1246762 

8.3491256 

001718213 

683 

339889 

198155287 

24.1453929 

8.3539047 

.001715266 

684 

341056 

199176704 

24.1660919 

8.3586784 

.001712329 

686 

342225 

200201625. 

24.1867732 

8.3634466 

.001709402 

686 

343396 

201230056 

24.2074369 

8.3682095 

.001706485 

687 

344569 

202262003 

24.2280829 

8.3729668 

.001703578 

683 

345744 

203297472 

24.2487113 

8.3777188 

.001700680 

689 

346921 

204336469 

24.2693222 

8.3824653 

.001697793 

690 

348100 

205379000 

24.2899156 

8.3872065 

.001694916 

691 

349281 

206425071 

24.3104916 

8.3919423 

.001692047 

592 

350464 

207474688 

24.3310501 

8.3966729 

.001689189 

593 

351649 

203527857 

24.3515913 

8.4013981 

.001686341 

594 

352836 

209584584 

24.3721152 

8.4061180 

.001683502 

595 

354025 

210644875 

24.3926218 

8.4108326 

.001680672 

596 

355216 

211708736 

24.4131112 

8.4155419 

.001677852 

597 

356409 

212776173 

24.4335834 

8.4202460 

.001675042 

598 

357604 

213847192 

24.4540385 

8.4249448 

.001672241 

599 

358801 

214921799 

24.4744765 

8.4296383 

.001669449 

600 

360000 

216000000 

24.4948974 

8.4343267 

.001666667 

601 

361201 

217081801 

24.5153013 

8.4390098 

.001663894 

602 

362404 

218167208 

24.5356883 

8.4436877 

.001661130 

603 

363609 

219256227 

24.5560583 

8.4483605 

.001658375 

604 

364816 

220348864 

24.5764115 

8.4530281 

.001655629 

605 

366025 

221-445125 

24.5967478 

8.4576906 

.001652893 

606 

367236 

222545016 

24.6170673 

8.4623479 

.001650165 

607 

363449 

223648543 

24.6373700 

8.4670001 

.001647446 

608 

369664 

224755712 

24.6576560 

8.4716471 

.001644737 

609 

370881 

225866529 

24.6779254 

8.4762892 

.001642036 

610 

372100 

226981000 

24.6981781 

8.4809261 

.001639344 

611 

373321 

22S099131 

24.7184142 

8.4855579 

001636661 

612 

374544 

229220928 

24.7386338 

8.4901848 

.001633987 

613 

375769 

230346397 

24.7588368 

8.4948065 

.001631321 

614 

376996 

231475544 

24.7790234 

8.4994233 

.001628664 

615 

378225 

232608375 

24.7991935 

8.5040350 

.001626016 

616 

379456 

233744396 

24.8193473 

8.5086417 

.001623377 

617 

380639 

2348S5I13 

24.8394847 

8.5132435 

001620746 

618 

381924 

236029032 

24.8596058 

8.5178403 

001618123 

619 

383161 

237176659 

24.8797106 

8.5224321    001615509 

620 

334400 

238328000 

24.8997992 

8.5270139    .001612303 

214          TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS, 


1  «* 

Squares. 

Cubes. 

Square  Boots 

Cube  Roots. 

Reciprocals. 

621 

385641 

239483061 

24.9198716 

8.5316009 

.001610306 

622 

386884 

240641848 

24.9399278 

8.5361780 

.001607717 

623 

388129 

241804367 

24.9599679 

8.5407501 

.001605136 

624 

389376 

242970624 

24.9799920 

8.6453173 

.001602504 

625 

390625 

244140625 

25.0000000 

8.5498797 

.001600000 

626 

391876 

245314376 

25.0199920 

8.6544372 

.001597444 

627 

393129 

246491883 

25.0399681 

8.5589899 

.001594896 

628 

394384 

247673152 

25.0599282 

8.6635377 

.001592357 

629 

395641 

243858189 

25.0798724 

8.5680807 

.001589825 

630 

396900 

250047000 

25.0998008 

8.5726189 

.001687302 

631 

398161 

251239591 

25.1197134 

8.5771523 

.001584780 

632 

399424 

252435968 

25.1396102 

8.5816809 

.001582278 

633 

400689 

253636137 

25.1594913 

8.5862047 

.001579779 

634 

401956 

254840104 

25.1793566 

8.5907238 

.001677287 

635 

403225 

256047875 

25.1992063 

8.5952380 

.001574803 

636 

404496 

257259456 

25.21904(^4 

8.6997476 

.001572327 

637 

405769 

258474853 

25.2388589 

8  6042525 

.001569859 

633 

407044 

259694072 

25.2586619 

8.6087526 

.001567398 

639 

408321 

260917119 

25.2784493 

8.6132480 

.001564946 

640 

409600 

262144000 

25.2932213 

8.6177388 

.001562500 

641 

410881 

263374721 

25.3179778 

8.6222248 

.001560062 

642 

412164 

264609233 

25.3377189 

8.6267063 

.001557632 

643 

413449 

265847707 

25.3574447 

8.6311830 

.001555210 

644 

414736 

267089934 

25.3771551 

8.6356551 

.001552796 

645 

416025 

268336125 

25.3968502 

8.6401226 

.001550388 

646 

417316 

269586136 

25.4165301 

8.6445855 

.001547988 

647 

418609 

270840023 

25.4361947 

8.6490437 

.001645595 

643 

419904 

272097792 

25.4558441 

8.6534974 

.001543210 

649 

421201 

273359449 

25.4754784 

8.6579465 

.001540832 

650 

422500 

274625000 

25.4950976 

8.6623911 

.001538462 

651 

423801 

275894451 

25.5147016 

8.6668310 

.001536098 

652 

425104 

277167808 

25.5342907 

8.6712665 

.001533742 

653 

426409 

278445077 

25.5538647 

8.6756974 

.001531394 

654 

427716 

279726264 

'  25.5734237 

8.6801237 

.001529052 

655 

429025 

281011375 

25.5929678 

8.6845456 

.001526718 

656 

430336 

282300416 

25.6124969 

8.6889630 

.001524390 

657 

431649 

283593393 

25.6320112 

8.6933759 

.001522070 

658 

432964 

284890312 

25.6515107 

8.6977843 

.001519757 

659 

434231 

286191179 

25.6709953 

8.7021882 

.001517451 

660 

435600 

287496000 

25.6904652 

8.7065877 

.001515152 

661 

436921 

288804781 

25.7099203 

8.7109827 

.001512869 

662 

438244 

290117528 

25.7293607 

8.7153734 

.001510574 

663 

439569 

291434247 

25.7487864 

8.7197590 

.001508296 

664 

440896 

292754941 

25.7681975 

8.7241414 

.001506024 

665 

442225 

294079625 

25.7875939 

8.7285187 

.001503769 

666 

443556 

295408296 

25.8069758 

8.7328918 

.001501502 

667 

444889 

296740963 

25.8263431 

8.7372604 

.001499250 

668 

446224 

298077632 

25.8456960 

8.7416246 

.001497006 

669 

447561 

299418309 

25.8650343 

8.7459846 

.001494768 

670 

448900 

300763000 

25.8843582 

8.7503401 

.001492537 

671 

450241 

302111711 

25.9036677 

8.7546913 

.001490313 

672 

451584 

303464448 

25.9229628 

8.7590383 

.001488095 

673 

452929 

304821217 

25.9422435 

8.7633809 

.001485384 

674 

454276 

306182024 

25.9615100 

8.7677192 

.001483680 

675 

455625 

307546875 

25.9807621 

8.7720532 

.001481481 

676 

456976 

308915776 

26.0000000 

8.7763830 

.001479290 

677 

458329 

310288733 

26.0192237 

8.7807084 

.001477105 

678 

459684 

311665752 

26.0384331 

8.7850296 

.001474926 

679 

461041 

313046839 

26.0576284 

8.7893466 

.001472754 

680 

462400 

314432000 

26.0763096 

8.7936593 

.001470588 

681 

463761 

315821241 

26.0959767 

8.7979679 

.001468429 

682 

465124 

317214568 

26.1151297 

8.8022721 

.001466276 

CUBE   ROOTS,    AND   RECIPROCALS. 


215 


Wo. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocate. 

683 

466489 

318611987 

26.  1342687 

8.8065722 

.001464129 

684 

467856 

320013504 

26.1533937 

8.8108681 

.001461988 

685 

469225 

321419125 

26.1725047 

8.8151598 

.001459854 

686 

470596 

322828856 

26.1916017 

8.8194474 

.001457726 

687 

471969 

324242703 

26.2106848 

8.8237307 

.001455604 

688 

473344 

325660672 

26.2297541 

8.8280099 

.001453488 

689 

474721 

327082769 

26.2488095 

8.8322850 

.001451379 

690 

476100 

328509000 

26.2678511 

8.8365559 

.001449276 

691 

477481 

329939371 

26.2868789 

8.8408227 

.001447178 

692 

478864 

331373888 

26.3058929 

8.8450854 

.001445087 

693 

480249 

332812557 

26.3248932 

8.8493440 

.001443001 

694 

481636 

334255384 

26.3438797 

8.8535985 

.001440922 

695 

483025 

335702375 

26.3628527 

8.8578489 

.001438849 

696 

484416 

337153536 

26.3818119 

8.8620952 

.001436782 

697 

485809 

338608873 

26.4007576 

8.8663375 

.001434720 

693 

487204 

340068392 

26.4196896 

8.8705757 

.001432665 

699 

488601 

341532099 

26.4386081 

8.8748099 

.001430615 

700 

490000 

343000000 

26.4575131 

8.8790400 

.001428571 

701 

491401 

344472101 

26.4764046 

8.8832661 

.001426534 

702 

492804 

345948408 

26.4952826 

8.8874882 

.001424501 

703 

494209 

347428927 

26.5141472 

8.8917063 

.001422476 

704 

495616 

348913664 

26.5329983 

8.8959204 

.001420455 

705 

497025 

350402625 

26.5518361 

8.9001304 

.001418440 

706 

498436 

351895816 

26.5706605 

8.9043366 

.001416431 

707 

499849 

353393243 

26.5894716 

8.9085387 

.001414427 

708 

501264 

354894912 

26.6082694 

8.9127369 

.001412429 

709 

502681 

356400829 

26.6270539 

8.9169311 

.001410437 

710 

504100 

357911000 

26.6458252 

8.9211214 

.001408451 

711 

505521 

359425431 

26.16645833 

8.9253078 

.001406470 

712 

506944 

360944128 

26.6833281 

8.9294902 

.001404494 

713 

508369 

362467097 

26.7020593 

8.9336687 

.001402525 

714 

509796 

363994344 

26.7207784 

8.9378433 

.001400560 

715 

511225 

365525875 

26.7394839 

8.9420140 

.001398601 

716 

512656 

367061696 

26.7581763 

8.9461809 

.001396648 

717 

514089 

368601813 

26.7768557 

8.9503438 

.001394700 

718 

515524 

370146232 

26.7955220 

8.9545029 

001392768 

719 

516961 

371694959 

26.8141754 

8.9586581 

.001390821 

720 

518400 

373248000 

26.8328157 

8.9628095 

.001388889 

721 

519341 

374805361 

26.8514432 

8.9669570 

.001386963 

722 

621284 

376367048 

26.8700577 

8.9711007 

.001386042 

723 

522729 

377933067 

26.8886593 

8.9752406 

.001383126 

724 

524176 

379503424 

26.9072481 

8.9793766 

.001381215 

725 

525625 

381078125 

26.9258240 

8.9835089 

.001379310 

726 

527076 

382657176 

26.9443872 

8.9876373 

.001377410 

727 

528529 

384240583 

26.9629375 

8.9917620 

.001375516 

728 

529984 

385828352 

26.9814751 

8.9958829 

.001373626 

729 

531441 

387420489 

27.0000000 

9.0000000 

.001371742 

730 

632900 

389017000 

27.0185122 

9.0041134 

.001369863 

731 

634361 

390617891 

27.0370117 

9.0082229 

.001367989 

732 

535824 

392223168 

27.0554985 

9.0123288 

.001366120 

733 

537289 

393832837 

27.0739727 

9,0164309 

.001364256 

734 

638756 

395446904 

27.0924344 

9.0205293 

.001362398 

735 

540225 

397065375 

27.1108834 

9.0246239 

.001360544 

736 

541696 

39S68S256 

27.1293199 

9.0287149 

.001358696 

737 

543169 

400315553 

27.1477439 

9.0328021 

.001356852 

738 

544644 

401947272 

27.1661554 

9.0368857 

.001355014 

739 

546121 

403583419 

27.1845544 

9.0409655 

.001353180 

740 
741 
742 
i   743 
744 

547600 
549081 
550564 
552049 
553536 

405224000 
406869021 
408518488 
410172407 
411830784 

27.2029410 
27.2213152 
27.2396769 
27.25R0263 
272763634 

9.0450417 
9.0491142 
9,0531831 

9.0572482 
9.0613098 

.001351351 
.001349528 
.001347709 
.001345895 
.001344086 

216          TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS, 


No. 

Sqaun*. 

Cubes. 

Square  Roots. 

Gale  Roots. 

Reciprocals. 

745 

555025 

413493625 

27.2946881 

9.0653677 

.001342282 

746 

556516 

415160936 

27.3130006 

9.0694220 

.001340483 

747 

553009 

416832723 

27.3313007 

9.0734726 

.001338688 

748 

559504 

418508992 

27.3495887 

9.0776197 

.001336898 

749 

561001 

420189749 

27.3678644 

9.0815631 

.001335113 

750 

562500 

421875000 

27.3861279 

9.0856030 

.001333333 

751 

564001 

423564751 

27.4043792 

9.0896392 

.001331558 

752 

565504 

425259008 

27.4226184 

9.0936719 

.001329787 

753 

567009 

426957777 

27.4408455 

9.0977010 

.001328021 

754 

568516 

428661064 

27.4590604 

9.1017265 

.001326260 

755 

570025 

430368875 

27.4772633 

9.1057485 

.001324503 

756 

571536 

432081216 

27.4954542 

9.1097669 

.001322751 

757 

573049 

433798093 

27.5136330 

9.1137818 

.001321004 

758 

574564 

435519512 

27.5317998 

9.1177931 

.001319261 

759 

576081 

437245479 

27.5499546 

9.1218010 

.001317523 

760 

577600 

438976000 

27.5680975 

9.1258053 

.001315789 

761 

579121 

440711081 

27.5862284 

9.1298061 

.001314060 

762 

580644 

442450728 

27.6043475 

9.1338034 

.001312336 

763 

582169 

444194947 

27.6224546 

9.1377971 

.001310616 

764 

583696 

445943744 

27.6405499 

9.1417874 

.001308901 

765 

585225 

447697125 

27.6586334 

9.1457742 

.001307190 

766 

586756 

449455096 

27.6767050 

9.1497576 

.001305483 

767 

588289 

451217663 

27.6947648 

9.1537375 

.001303781 

768 

589824 

452984832 

27.7128129 

9.1577139 

.001302083 

769 

591361 

454756609 

27.7308492 

9.1616869 

.001300390 

770 

592900 

456533000 

27.7488739 

9.1656565 

.001298701 

771 

594441 

458314011 

27.7668868 

9.1696225 

.001297017 

772 

595984 

460099648 

27.7848880 

9.1735852 

.001295337 

773 

597529 

461889917 

27.8028775 

9.1775445 

.001293661 

774 

599076 

463684824 

27.8208555 

9.1815003 

.001291990 

775 

600625 

465484375 

27.8388218 

9.1854527 

.001290323 

778 

602176 

467288576 

27.8567766 

9.1894018 

.001288660 

777 

603729 

469097433 

27.8747197 

9.1933474 

.001287001 

778 

605284 

470910952 

27.8926514 

9.1972897 

.001285347 

779 

606841 

472729139 

27.9105715 

9.2012286 

.001283697 

780 

608400 

474552000 

27.9284801 

9.2051641 

.001282051 

781 

609961 

476379541 

27.9463772 

9.2090962 

.001280410 

782 

611524 

478211768 

27.3642629 

9.2130250 

.001278772 

783 

613039 

480048687 

27.9821372 

9.2169505 

.001277139 

784 

614656 

481890304 

28.0000000 

9.2208726 

.001275510 

785 

616225 

483736625 

28.0178515 

9.2247914 

.001273885 

786 

617796 

485587656 

28.0356915 

9.2287068 

.001272265 

787 

619369 

487443403 

28.0535203 

9.2326189 

.001270648 

788 

620944 

489303872 

28.0713377 

9.2365277 

.001269036 

789 

622521 

491169069 

28.0891438 

9.2404333 

.001267427 

790 

624100 

493039000 

28.1069386 

9.2443355 

.001265823 

1   791 

625681 

494913671 

28.1247222 

9.2482344 

.001264223 

792 

627264 

496793088 

28.1424946 

9.25213QP 

.001262626 

793 

628849 

498677257 

28.1602557 

9.2560224 

.001261034 

794 

630436 

500566184 

28.1780056 

9.2599114 

.001259446 

795 

632025 

502459875 

28.1957444 

9.2637973 

.001257862 

796 

633616 

504358a36 

28.2134720 

9.2676798 

.001256281 

797 

635209 

506261573 

28.2311884 

9.2715592 

.001254705 

798 

636804 

508169592 

28.2488938 

9.2754352 

.001253133 

799 

638401 

510082399 

28.2665881 

9.2793081 

.001251564 

800 

640000 

512000000 

28.2842712 

9.2831777 

.001250000 

801 

641601 

513922401 

28.3019434 

9.2870440 

.001248439 

802 

643204 

515849608 

28.3196045 

9.2909072 

.001246883 

803 

644809 

517781627 

28.3372546 

9.2947671 

,001245330 

804 

646416 

519718464 

28.3548938 

9.2986239 

.001243781 

805 

648025 

521660125 

23.3725219 

9.3024775 

.001242236 

806 

649636 

523606616 

28.3901391 

9.3063278 

.001240695 

CUBE    ROOTS,    AND    RECIPROCALS. 


217 


No. 

Squares. 

Cubes. 

Square  Boots. 

Cube  Roots. 

Reciprocals. 

807 

651249 

525557943 

28.4077454 

9.3101750 

.001239167 

808 

6528*4 

5275141  12 

28.4253408 

9.3140190 

.001237624 

809 

654481 

529475129 

28.4429253 

9.3178599 

.001236094 

810 

656100 

531441000 

28.4604989 

9.3216975 

.001234568 

811 

657721 

533411731 

28.4780617 

9.3255320 

.001233046 

812 

659344 

535387328 

28.4956137 

9.3293634 

.001231527 

813 

660969 

537367797 

28.5131549 

9.3331916 

.001230012 

814 

662596 

539353144 

28.5306852 

9.3370167 

.001228501 

815 

664225 

541343375 

28.5482048 

9.3408386 

.001226994 

816 

665856 

543338496 

28.5657137 

9.3446575 

.001225490 

817 

667489 

545338513 

28.5832119 

9.3484731 

.001223990 

818 

669124 

547343432 

28.6006993 

9.3522857 

.001222494 

819 

670761 

649353259 

286181760 

9.3560952 

.001221001 

820 

672400 

65136800C 

28.6356421 

9.3599016 

.001219512 

821 

674041 

553387661 

28.6530976 

9.3637049 

.001218027 

822 

675684 

555412248 

28.6705424 

9.3675051 

.001216545 

823 

677329 

557441767 

28.6379766 

9.3713022 

.001215067 

824 

678976 

559476224 

28.7054002 

9.3750963 

.001213592 

825 

6S0625 

561515625 

28.7228132 

9.3788873 

.001212121 

826 

682276 

563559976 

23.7402157 

9.3826752 

.001210654 

827 

683929 

565609283 

23.7576077 

9.3864600 

.001209190 

828 

685584 

567663552 

28.7749891 

9.3902419 

.001207729 

829 

687241 

569722789 

28.7923601 

9.3940206 

.001206273 

830 

688900 

571787000 

28.8097206 

9.3977964 

.001204819 

831 

690561 

573856191 

28.8270706 

9.4015691 

.001203369 

832 

692224 

5759303S8 

28.8444102 

9.4053387 

.001201923 

833 

693889 

578009537 

28.8617394 

9.4091054 

.001200480 

834 

695556 

580093704 

23.8790582 

9.4128690 

.001199041 

835 

697225 

582182375 

28.8963666 

9.4166297 

.001197605 

836 

698896 

684277056 

28.9136646 

9.4203873 

.001196172 

837 

700569 

586376253 

28.9309523 

9.4241420 

.001194743 

838 

702244 

588480472 

28.9482297 

9.4278936 

.001193317 

839 

703921 

590589719 

28.9654967 

9.4316423 

.001191895 

840 

705600 

592704000 

28.9827535 

9.4353880 

.001190476 

841 

707281 

594823321 

29.0000000 

9.4391307 

.001189061 

842 

708964 

5969476.38 

29.0172363 

9.4428704 

.001187648 

843 

710649 

599077107 

29.0344623 

9.4466072 

.001186240 

844 

712336 

601211584 

29.0516781 

9.4503410 

.001184834 

845 

714025 

603351125 

29.068-8837 

9.4540719 

.001183432 

846 

715716 

605495736 

29.0860791 

9.4577999 

.001182033 

847 

717409 

607645423 

29.1032644 

9.4615249 

.001180638 

848 

719104 

609800192 

29.1204396 

9.4652470 

.001179245 

849 

720801 

611960049 

29.1376046 

9.4639661 

.001177856 

850 

722500 

614125000 

29.1547595 

9.4726824 

.001176471 

851 

724201 

616295051 

29.1719043 

9.4763957 

.001175088 

852 

725904 

618470208 

29.  1890390 

9.4801061 

.001173709 

353 

727609 

,620650477 

29.2061637 

9.4838136 

.001172333 

854 

729316 

622835364 

29.2232784 

9.4875182 

.001170960 

855 

731025 

625026375 

29.2403330 

9.4912200 

.001169591 

856 

732736 

627222016 

29.2574777 

9.4949188 

.001168224 

857 

734449 

629422793 

29.2745623 

9.4986147 

.001166861 

853 

736164 

631623712 

29.2916370 

9.5023078 

.001165501 

859 

737881 

633839779 

29.3087018 

9.5059980 

.001164144 

860 

739600 

636056000 

29.3257566 

9.5096354 

.001  162791 

861 

741321 

638277331 

29.3423015 

9.5133699 

.001161440 

862 

743044 

640503923 

29.3593365 

9.5170515 

.001160093 

863 

744769 

642735647 

29.3763616 

9.5207303 

.001158749 

864 

746496 

644972544 

29.3933769 

9.5244063 

.001157407 

865 

748225 

647214625 

29.4103823 

9.5230794 

001156069 

866 

749956 

649161896 

29.4278779 

9.5317497 

.001154734 

867 

751689 

631714363 

29.4443637 

9.5354172 

.001153403 

863 

753424 

653972032 

29.4618397 

9.5390818 

.001152074 

TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS, 


No. 

Square*. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocate 

869 

755161 

656234909 

29.4788059 

9.5427437 

.001150748 

870 

766900 

658503000 

29.4957624 

9.5464027 

.001149426 

871 

758641 

660776311 

29.5127091 

9.6500589 

.001148106 

872 

760384 

663054848 

29.5296461 

9.5537123 

.001146789 

873 

762129 

665338617 

29.5465734 

9.5573630 

.001  145475 

874 

763876 

667627624 

29.5634910 

9.5610108 

.001144166 

875 

765625 

669921875 

29.5803989 

9.5646559 

.001142867 

876 

767376 

672221376 

29.5972972 

9.5682982 

.001141553 

877 

769129 

674526133 

29.6141858 

9.5719377 

.001140251 

878 

770884 

676836152 

29.6310648 

9.6755746 

.001138952 

879 

772641 

679151439 

29.6479342 

9.5792085 

.001137656 

880 

774400 

681472000 

29.6647939 

9.5828397 

.001136364 

881 

776161 

683797841 

29.6816442 

9.5864682 

.001135074 

882 

777924 

686128968 

29.6984848 

9.5900939 

.001133787 

883 

779689 

688465387 

29.7153159 

9.5937169 

.001132503 

884 

781456 

690807104 

29.7321375 

9.5973373 

.001131222 

886 

783225 

693154125 

29.7489496 

9.6009548 

.001129944 

886 

784996 

695506456 

29.7657521 

9.6045696 

.001128668 

887 

786769 

697864103 

29.7825452 

9.6081817 

.001127396 

888 

788544 

700227072 

29.7993289 

9.6117911 

.001126126 

889 

790321 

702595369 

29.8161030 

9.6153977 

.001124859 

890 

792100 

704969000 

29.8328678 

9.6190017 

.001123596 

891 

793881 

707347971 

29.8496231 

9.6226030 

.001122334 

892 

795664 

709732288 

29.8663690 

9.6262016 

.001121076 

893 

797449 

712121957 

29.8831056 

9.6297975 

.001119821 

894 
896 

799236 
801025 

714516984 
716917375 

29.8998328 
29.9165506 

9.6333907 
9.6369812 

.001118668 
.001117318 

896 

802816 

719323136 

29.9332591 

9.6405690 

.001116071 

897 
898 

804609 
806404 

721734273 
724150792 

29.9499583 
29.9666481 

9.6441542 
9.6477367 

.001114827 
.001113586 

899 

808201 

726572699 

29.9833287 

9.6513166 

.001112347 

900 

810000 

729000000 

30.0000000 

9.6548938 

.001111111 

901 
902 
903 

811801 
813604 
816409 

731432701 
733870808 
736314327 

30.0166620 
30.0333148 
30.0499584 

9.6584684 
9.6620403 
9.6656096 

.001109878 
.001108647 
.001107420 

904 

817216 

738763264 

30.0665928 

9.6691762 

.001106195 

906 

819025 

741217625 

30.0832179 

9.6727403 

.001104972 

906 

820836 

743677416 

30.0998339 

9.6763017 

.001103753 

907 

322649 

746142643 

30.1164407 

9.6798604 

.001102536 

908 

824464 

748613312 

30.1330383 

9.6834166 

.001101322   I 

909 

826281 

751089429 

30.1496269 

9.6869701 

.001100110 

910 

828100 

753571000 

30.1662063 

9.6905211 

.001098901 

911 

829921 

756058031 

30.1827765 

9.6940694 

.001097695 

912 

831744 

758550528 

30.1993377 

9.6976151 

.001096491 

913 

833569 

761048497 

30.2158899 

9.7011583 

.001095290 

914 

835396 

763551944 

30.2324329 

9.7046989 

.001094092 

915 

837225 

766060875 

30.2489669 

9.7082369 

.001092896 

916 

839056 

768575296 

30.2654919 

9.7117723 

.001091703 

917 

840889 

771095213 

30.2820079 

9.71530C1 

.001090513 

918 

842724 

773620632 

30.2985148 

9.7188354 

.001089325 

919 

844561 

776151559 

30.3150128 

9.7223631 

.001088139 

920 
921 

846400 
848241 

778688000 
781229961 

30.3315018 
30.3479818 

9.7258883 
9.7294109 

.001086957 
.001085776 

922 
923 

850084 
851929 

783777448 
786330467 

30.3644529 
30.3809151 

9.7329309 
9.7364484 

.001084599 
.001083424 

924 

853776 

788889024 

30.3973683 

9.7399634 

.001082251 

925 

855625 

791453125 

30.4138127 

9.7434758 

.001081081 

926 

857476 

794022776 

30.4302481 

9.7469857 

.001079914 

927 
928 

859329 
861184 

796597983 
799178752 

30.4466747 
30.4630924 

9.7504930 
9.7539979 

.001078749 
.001077586 

929 

863041 

801765089 

30.4795013 

9.7575002 

.001076426 

930 

864900  '   804357000 

30.4959014 

9.7610001 

.001075269 

CUBE   HOOTS,    AND   RECIPROCALS. 


219 


INo. 

Squares. 

Cubes. 

Squaro  Roots. 

Cube  Roots. 

Reciprocals. 

931 

866761 

806954491 

30.5122926 

9.7644974 

.001074114 

932 

86S624 

809557568 

30.5286750 

9.7679922 

001072961 

933 

870489 

812166237 

30.5450487 

9.7714845 

001071811 

934 

872356 

814780504 

30.5614136 

9.7749743 

.001070664 

935 

874225 

817400375 

30.5777697 

9.7781616 

.001069519 

936 

876096 

820025856 

30.5941171 

9.781-J466 

.001063376 

937 

877969 

822656953 

30.6104557 

9.7854238 

.001067236 

933 

879844 

825293672 

30.6267857 

9.7389037 

.001066098 

939 

881721 

827936019 

30.6431069 

9.7923361 

.001064963 

940 

883600 

830584000 

30.6594194 

9.7958611 

.001063830 

941 

885481 

833237621 

30.6757233 

9.7993336 

.001062699 

942 

837364 

835896388 

30.6920185 

9.8023036 

.001061571 

943 

839249 

838561807 

30.7083051 

9.8062711 

.001060445 

944 

891136 

841232384 

30.7245830 

9.8097362 

.001059322 

945 

893025 

843903625 

30.7408523 

9.8131989 

.001058201 

946 

894916 

346590536 

30.7571130 

9.8166591 

.001057082 

947 

896809 

849278123 

30.7733651 

9  8201  169 

.001055966 

94S 

898704 

851971392 

30.7896086 

9.8235723 

.001054852 

949 

900601 

854670349 

30.8058436 

9.8270252 

.001053741 

950 

902500 

857375000 

30.8220700 

9.8304757 

.001052632 

951 

904401 

860085351 

30.8332879 

9.8339238 

.001051525 

952 

906304 

862801408 

30.8544972 

9.8373695 

.001050420 

953 

908209 

865523177 

30.8706981 

9.8408127 

.001049318 

954 

910116 

868250664 

30.8868904 

9.8442536 

.001043218 

955 

912025 

870983875 

30.9030743 

9.8476920 

.001047120 

956 

913936 

873722816 

30.9192497 

9.8511280 

.001046025 

957 

915849 

876467493 

30.9354166 

9.8545617 

.001044932 

958 

917764 

879217912 

30.9515751 

9.8579929 

.001043341 

959 

919631 

881974079 

30.9677251 

9.8614213 

.001042753 

960 

921600 

884736000 

30.9838668 

9.8643483 

.001041667 

961 

923521 

887503631 

31.0000000 

9.8682724 

.001040583 

962 

925444 

890277128 

31.0161248 

9.8716941 

.001039501 

963 

927369 

893056347 

31  0322413 

9.8751135 

.001038422 

964 

929296 

895841344 

31.0483494 

9.8785305 

.001037344 

965 

931225 

898632125 

31.0644491 

9.8819451 

.001036269 

966 

933156 

901423696 

31.0805405 

9.8853574 

.001035197 

967 

935089 

904231063 

31.0966236 

9.8887673 

.001034126 

968 

937024 

907039232 

31.1126984 

9.8921749 

.001033058 

969 

938961 

909853209 

31.1237648 

9.8955801 

.001031992 

970 

940900 

912673000 

31.1443230 

9.8989830 

.001030928 

971 

942841 

915498611 

31.1608729 

9.9023835 

.001029366 

972 

944784 

918330048 

31.1769145 

9.9057817 

.001028807 

973 

946729 

921167317 

31.1929479 

9.9091776 

.001027749 

974 

948676 

924010424 

31.2089731 

9.9125712 

.001026694 

975 

950625 

926359375 

31.2249900 

9.9159624 

.001025641 

976 

952576 

929714176 

31.2409987 

9.9193513 

.001024590 

977 

954529 

932574833 

31.2569992 

9.9227379 

.001023541 

978 

956484 

935441352 

31.2729915 

9.9261222 

.001022495 

979 

958441 

938313739 

31.2889757 

9.9295042 

.001021450 

980 

960400 

941192000 

31.3049517 

9.9323839 

.001020408 

931 

962361 

944076141 

31.3209195 

9.9362613 

.001019368 

982 

964324 

946%6168 

31.3368792 

9.9396363 

.001018330 

983 

966239 

949H62087 

31.3528308 

9.9430092 

.001017294 

934 

963256 

952763904 

31.3637743 

9.9463797 

.001016260 

935 

970225 

955671625 

31.3847097 

9.9497479 

.001015228 

988 

972196 

958585256 

31.4006369 

9.9531138 

.001014199 

987 

974169 

961504303 

31.4165561 

9.9564775 

.001013171 

938 

976144 

264430272 

31.4324673 

9.9598389 

.001012146 

989 

978121 

967361669 

31.4433704 

9.9631981 

.001011122 

990 

980100 

970299000 

31.4&42654 

9.9665549 

.001010101 

991 

982081 

973242271 

31.4801525 

9.9699095 

.001009082 

992 

934064 

976191488 

31.4960315 

9.9732619 

.001008065 

220      TABLE   XIII.       SQUARES,    CUBES,    SQUARE   ROOTS,    &C. 


No. 

Squaw*. 

Cubes. 

Square  Boots. 

Cube  Boots. 

"Reciprocals. 

993 

986049 

979146657 

31  5119025 

9.9766120 

.001007049 

994 

988036 

982107784 

31.5277655 

9.9799599 

.001006036 

995 
996 

990025 
992016 

985074875 
988047936 

31.5436206 
31.5594677 

9.9833055 
9.9866488 

.001005025 
.001004016 

997 
993 

994009 
996004 

991026973 
994011992 

31.5753068 
31.5911380 

9.9899900 
9.9933289 

.001003009 
.001002004   ; 

999 

99S001 

997002999 

31.6061,613 

9.9966656 

001001001 

1000 
1001 
1002 
1003 
1004 
1005 
1006 
1007 
1008 
1009 

1000000 
1002001 
1004004 
1006009 
1008016 
1010025 
1012036 
1014049 
1016064 
1018081 

1000000000 
1003003001 
1006012003 
1009027027 
1012048064 
1015075125 
1018108216 
1021147343 
1024192512 
1027243729 

31.6227766 
31.6385.340 
31.6543836 
31.6701752 
31.6859590 
31.7017349 
31.7175030 
31.7332633 
31.7490157 
31.7647603 

10.0000000 
10.0033322 
10.0066622 
10.0099899 
10.0133155 
10.0166389 
10.0199601 
10.0232791 
10.0265958 
10.0299104 

.001000000 
.0009990010  ! 
.0009980040  i 
.0009970090 
.0009960159 
.0009950249 
.0009940358 
.0009930487 
.0009920635 
.0009910803 

1010 
1011 
1012 

1020100 
1022121 
1024144 

1030301000 
1033364331 
1036433728 

31.7804972 
31.7962262 
31.8119474 

10.0332228 
10.0365330 
10.0398410 

.0009900990 
.0009891197 
.0009881423 

1013 
1014 
1015 

1026169 
1028196 
1030225 

1039509197 
1042590744 
1045678375 

31.8276609 
31.8433666 
31.8590646 

10.0431469 
10.0-464506 
10.0497521 

.0009871668 
.0009861933 
.0009852217 

1016 
1017 

1032256 
1034239 

1048772096 
1051871913 

31.8747549 
31.8904374 

10.0530514 
10.0563485 

.0009842520 
.0009832842 

1018 
1019 

1036324 
1038361 

1054977832 
1058089859 

31.9061123 
31.9217794 

10.0596435 
10.0629364 

.0009823183 
.0009813543 

1020 
1031 
1022 
1023 

1040400 
J042441 
1044484 
1046529 

1061208000 
1064332261 
1067462648 
1070599167 

31  9374388 
31.9530906 
31.9687347 
31.9843712 

10.0662271 
10.0695156 
10.0728020 
10.0760863 

.0009803922 
.0009794319 
.0009784736 
.0009775171 

1024 
1025 
1026 
1027 

1048576 
1050625 
105267G 
1054729 

1073741824 
1076890625 
1080045576 
1083206683 

32.0000000 
32.0156212 
32.0312348 
32.0468407 

10.0793684 
10.0826484 
10.0859262 
10.0892019 

.0009765625 
.0009756098 
.0009746589 
.0009737098 

1028 

1056784 

1086373952 

32.0624391 

10.0924755 

.0009727626 

1029 

1058841 

10^9547389 

32.0780298 

10.0957469 

.0009718173 

1030 
1031 

1060900 
1062961 

1092727000 
1095912791 

32.0936131 
32.1091887 

10.0990163 
10.1022835 

.0009708738 
.0009699321 

1032 
1033 
1034 
1035 
1036 
1037 
1038 
1039 

1065024 
1067089 
1069156 
1071225 
1073296 
1075369 
1077444 
1079521 

1099104768 
1102302937 
1105507304 
1108717875 
1111934656 
1115157653 
1118386872 
1121622319 

32.  1247568 
32.1403173 
32.1558704 
32.1714159 
32.1869539 
32.2024844 
32.2180074 
32.2335229 

10.1055487 
10.1088117 
10.1120726 
10.1153314 
10.1185882 
10.1218428 
10.1250953 
10.1283457 

.0009689922 
.0009680542  1 
.0009671180 
.0009661836 
.0009652510 
.0009643202 
.0009633911  1 
.0009624639 

1040 

1081600 

1124864000 

32.2490310 

10.1315941 

.0009615385 

1041 
1042 
1043 

1083681 
1085764 
1087849 

1128111921 
1131366088 
1134626507 

32.2645316 
32.2800248 
32.2955105 

10.1348403 
10.1380845 
10.1413266 

.0009606148 
.0009596929 
.0009587728 

1044 

1089936 

1137893184 

32.3109888 

10.1445667 

.0009578544 

1045 

1092025 

1141166125 

32.3264598 

10.1478047 

.0009569378 

1046 

1094116 

1144445336 

32.3419233 

10.1510406 

.0009560229 

1047 

1096209 

1147730823 

32.3573794 

10.1542744 

.0009551098 

1048 
1049 

1098304 
1100401 

1151022592 
1154320649 

32.3728281 
32.3882695 

10  15^5062 
10.1607359 

.0009541985 
.0009532888 

1050 
1051 
1052 

1102500 
1104601 
1106704 

1157625000 
1160935651 
1164252608 

32.4037035 
32.4191301 
32.4345495 

10.1639636 
10.1671893 
10.1704129 

.0009523810 
.0009514748 
.0009505703 

1053 
1054 

1108809 
1110916 

1167575877 
1170905464 

32.4499615 
32.4653662 

10.1736344 
10.1768539 

0009496676 
.0009487666 

TABLE     XIV. 

LOGARITHMS   OF   NUMBERS. 

FROM    1    TO    10,000. 


222 


TABLE   XIV.      LOGARITHMS   OF   NUMBERS. 


No 

.  0 

0    7 

8 

9 

Dlff. 

1U 

1 
S 

j 

1 

)00000( 
4321 
1  860C 
1  012837 
7032 
>02118£ 
I  5306 

)000434 
4751 
902t 
01325S 
7451 
021  60c 
571£ 

0008& 
518 
945 
01368( 
786^ 
0220  H 
612L 

300130] 
560i 
987t 
)01410( 
*  8284 
5  022426 
>  653c 

00173 
>  6035 
>  '01  030 
)  452 
870 
02284 
694 

00216 
646 
010724 
494 
911 
02325 
735 

00259 
689 
01114 
536 
953 
02366 
775 

00302 
732 
01157 
677 
994 
02407 
8164 

00346 
774 
01199 
619 
02036 
448 
857 

00389" 
817 
01241 
661 
02077 
489 
897 

~432 
428 
424 
420 
416 
412 

7 

i 

9384 
(033424 

978 
03382 

0301  9.f 
4227 

03060C 
4625 

031004 

502 

03140 
543 

03181 
583 

03221 
623C 

03261 
662 

03302 
702£ 

404  1 
400  ' 

' 

7426 

782 

8223  862f 

901 

941 

981 

04020 

04060 

04099 

397 

lie 
i 

^ 

5 
6 

041393 
5323 
9218 
053078 
6905 
060693 
4458 

04178 
571 
9606 
05346 
728 
06107 
483 

042182042576 
6105  6495 
9993  05038C 
053846J  4230 
7666  8046 
061452'061829 
5206  5580 

04296 

688 
05076 
461 
842 
06220 
595 

04336 
727 
06115 
499 
880 
06258 
632 

04375 
766 
05153 
537 

918 
06295 
669 

04414 
805 
051924 
576C 
956 
06333 
707 

04454 

844^ 
05230 
6142 
994S 
06370S 
7443 

04493 

883C 
052694 
6524 
060320 
4083 
781 

393 
390 
386 
383 
379 
376  ! 
373 

8 
9 

8186 
071882 
6547 

855 
07225 
591 

8928 
072617 
6276 

9298 
072985 
6640 

9668 
073352 
7004 

07003 
371 
7368 

07040 
408 
773 

07077 
445 
8094 

071145 

4816 
8457 

07151 
6182 

881 

370 
366 
363 

120 

1 

2 
4 
5 
6 

r 
fc 

9 

079181 
082785 
6360 
9905 
093422 
6910 
100371 
3804 
7210 
110590 

079543 
083144 
6716 
090258 
3772 
7257 
100715 
4146 
7549 
110926 

079904 
083503 
7071 
090611 
4122 
7604 
101059 
4487 
7888 
111263 

080266 
3861 
7426 
090963 
4471 
7951 
101403 
4828 
8227 
111599 

080626 
4219 
778 
091315 
4820 
8298 
101747 
5169 
8565 
111934 

08098 
457 
8136 
091667 
5169 
8644 
102091 
6510 
8903 
12270 

08134 
4934 
8490 
092018 
5518 
8990 
102434 
6851 
9241 
112605 

08170 
529 
884 
09237 
686 
933o 
102777 
619 
9579 
12940 

082067 
6647 
9198 
092721 
6215 
9681 
103119 
6631 
9916 
113276 

082426 
6004 
9552 
09307 
6562 
100026 
3462 
6871 
110253 
3609 

360 
367 
365 
362 
349 
346 
343 
341 
338 
336 

130 

2 
3 
4 

5 
6 
7 
8 
9 

113943 
7271 
120574 
3852 
7105 
130334 
3539 
6721 
9879 
143015 

114277 
7603 

120903 
4178 
7429 
30655 
3858 
7037 
40194 
3327 

14611 
7934 
21231 
4504 
7753 
30977 
4177 
7354 
40508 
3639 

114944 
8265 
121560 
4830 
8076 
131298 
4496 
7671 
140822 
3951 

115278 
8595 

121888 
5156 
8399 
131619 
4814 
7987 
141136 
4263 

16611 

8926 
22216 
5481 
8722 
31939 
5133 
8303 
41450 
4574 

15943 
9256 
22544 
5806 
9045 
32260 
5451 
8618 
41763 
4885 

16276 

9586 
22871 
6131 
9368 
32580 
5769 
8934 
42076 
5196 

116608 
9915 
23198 
6456 
9690 
32900 
6086 
9249 
42389 
5507 

116940 
20245 
3525 
6781 
30012 
3219 
6403 
9564 
42702 
5818 

333 
330 
328 

326 
323 
321 
318 
316 
314 
311 

140 
1 

146128 
9219 

152288 

46438 
9527 
52594 

46748 
9835 
52900 

147058 
150142 
3205 

147367 
150449 
3510 

47676 
50756 
3815 

47985 
51063 
4120 

48294 
51370 
4424 

48603 
51676 

4728 

48911 
51982 
5032 

309 
307 
305 

3 
4 
5 

5336 
8362 
161368 

5640 
8664 
61667 

5943 
8965 
61967 

6246 
9266 
162266 

6549 
9567 
162564 

6852 
9868 
62863 

7154 
60168 
3161 

7457 
60469 
3460 

7759 
60769 
3758 

8061 
61068 
4055 

303 
301 
299 

6 
7 
8 
9 

4353 
7317 
170262 
3186 

4650 
7613 
170555 

3478 

4947 
7908 
70848 
3769 

5244 
8203 
171141 
4060 

5541 
8497 
171434 
4351 

5838 
8792 
71726 
4641 

6134 
9086 
72019 
4932 

6430 
9380 
72311 
5222 

6726 
9674 
72603 
6512 

7022 
9968 
72895 
6802 

297 
295 
293 
291 

150 

176091 

176381 

76670 

176959 

77248 

77536 

77825 

78113 

78401 

78689 

289 

2 
3 
4 
•6 

181844 
4C91 
7521 
190332 

9264 
182129 
4975 

7803 
90612 

95521  9839 
82415!  182700 
5259  6542 
8084  8366 
90892  191171 

80126 
2985 
5825 

8647 
191451 

80413 
3270 
6108 

8928 
91730 

80699 
3555 
6391 
9209 
92010 

80986 
3839 
6674 
9490 
92289 

81272 
4123 
6956 
9771 
92567 

81558 
4407 
7239 
190051 
•2846 

287 
285 
283 
281 
279 

7 
8 
9 

3125 

5900 
8657 
Z01397  ' 

3403 
6176 
8932 
&1670 

3681 
6453 
9206 
01943 

3959   4237 
6729   7005 
9481   9755 
302216  202-188 

4514 
7281 
00029 
2761 

4792 
7556 
00303 
3033 

5069 
7832 
00577 
3305 

5346 

8107 
00850 
3577 

5623 

8382 
201124 

3848 

278 
276 
274  i 
272 

No.|  0 

1 

3  I  3 

4 

5 

6 

7 

8 

9 

Iff.  ' 

i. 

TABLE   XIV.       LOGARITHMS    OF   NUMBERS. 


No. 

O 

1 

3 

3    4 

5 

6 

7 

8 

9 

Biff. 

160 
1 

204120 
6326 

204391 
7096 

204663204934205204 
7365:  7634  1  7904 

205475 
8173 

205746 
8441 

206016 
8710 

206286 
8979 

206556 
9247 

271 

269 

2 

9515 

9733 

210051  210319.210586 

210853 

211121 

211388 

211654 

211921 

287 

8 

212183 

212454 

2720  2986|  3252 

3518 

3783 

4049 

4314 

4579 

266 

4 

4344 

5109 

5373  5633,  5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7484 

7747 

8010  8273  8536 

8798 

9060 

9323 

9585 

9346 

262 

0 

220103 

220370 

220631220892221153 

221414 

221675 

221936 

222196 

222456 

261 

7 

2716 

2976 

3236  3496  3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309 

5568 

5326 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

9 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9632 

9933 

230193 

256 

170 

230449 

230704 

230960 

231215 

231470 

231724 

231979 

232234 

232488 

232742 

255 

1 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

2 

5523 

5781 

6033 

6235 

6537 

6789 

7041 

7292 

7544 

7795 

252 

3 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

240050 

240300 

250 

4 

240549 

240799 

241048 

241297 

241546 

241795 

242044 

242293 

2541 

2790 

249 

5 

3033 

3286 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248 

6 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

8954 

9193 

9443 

9687 

9932 

250176 

245 

8 

250420 

250664 

250903 

251151 

251395 

251633 

251881 

252125 

252363 

2610 

243 

9 

2853 

3096 

3383 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

255273 

255514 

255765 

255996 

256237 

256477 

256713 

256958 

257198 

257439 

241 

1 

7679 

7918 

8153 

8398 

8637 

8377 

9116 

9355 

9594 

9833 

239 

2 

260071 

260310 

260548 

260787 

261025 

261263 

261501 

261739 

261976 

262214 

238 

3 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

4 

4318 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

6 

9513 

9746 

9980 

270213 

270446 

270679 

270912 

2Z1144 

271377 

271609 

233 

7 

271842 

272074 

272306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

232 

8 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

6772 

6002 

6232 

230 

9 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

278754 

278982 

279211 

279439 

279667 

279895 

280123 

280351 

280578 

280806 

228 

1 

281033 

281261 

281488 

281715 

231942 

282169 

2396 

2622 

2349 

3075 

227 

9 

3301 

3527 

3753 

3979  4205 

4431 

4656 

4882 

5107 

5332 

226 

3 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9312 

223 

5 

290035 

290257 

290480 

290702 

290925 

291147 

291369 

291591 

291813 

292034 

222 

6 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221 

7 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

8 

6665 

6334 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

9 

8353 

9071 

9289 

9507 

9725 

'  9943 

300161 

300378 

300595 

300813 

218 

200 

301030 

301247 

301464 

301681 

301898 

302114 

302331 

302547 

302764 

302980 

217 

1 

3196 

3412 

3623 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

3 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

4 

9630 

9843 

310056 

310268 

310481 

310693 

310906 

311118 

311330 

311542 

212 

S 

311754 

311966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

8 

3887 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

7 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

8 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

320354 

320562 

320769 

320977 

321184 

321391 

321593 

321805 

322012 

207 

210 

322219 

322426 

322633 

322339 

323046 

323252 

323458 

323665 

323871 

324077 

206 

1 

4232 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

2 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 
4 

8380 
330414 

8583 
330617 

8787 
330819 

8991 
331022 

9194 
331225 

9398 
331427 

9601 
331630 

9805 
331832 

330008 
2034 

330211 
2236 

203 
202 

6 

2433 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

425? 

202 

6 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201 

7 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7853 

8058 

8257 

200 

8 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9349 

340047 

340246 

199 

9 

340444 

310642 

340341 

341039 

341237 

341435 

341632 

341830 

2028 

2225 

198 

No. 

0 

1 

3 

3 

4 

5 

6 

7 

6 

9 

DH. 

224, 


TABLE    XiV.       LOGARITHMS    OF    NUMUEI.'S. 


No. 

0 

1 

3 

343999 

Diff. 

220 

342423 

342620 

342817 

343014  343212 

343409 

343606 

343802 

344196 

197 

1 

4392 

4589 

4785 

4981   5178 

5374 

5570 

5766 

5962 

6157 

196 

2 

6353 

6549 

6744 

6939   7135 

7330 

7525 

7720 

7915 

8110 

195 

3 

8305 

8500 

8691 

8889   9083 

927S 

9472 

9666 

9860 

350054 

194 

4 

350248 

350442 

350636  1350829  351023 

351216 

351410 

351603 

351796 

1989 

193 

5 

2183 

2375 

2568 

2761   2954 

3147 

3339 

3532 

3724 

3916 

193 

6 

4108 

4301 

4493 

4685   4876 

5068 

5260 

5452 

5643 

5834 

192 

7 

6026 

6217 

64(,3 

6599   6790 

6981 

7172 

7363 

7554 

7744 

191 

8 

7935 

8125 

83]  6 

850C   8696 

8886 

9076 

9266 

9456 

9646 

190 

9 

9835 

360025 

360215 

360404 

360593 

360783 

360972 

361161 

361350 

361539 

189 

230 

361728 

361917 

J62105 

362294 

362482 

362671 

362859 

363048 

363236 

363424 

188 

1 

3612 

3800 

3988 

4176  4363 

4551 

4739 

4926 

5113 

6301 

188 

2 

6488 

5675 

5862 

6049   6236 

6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915!  8101 

8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772!  9958 

370143 

370328 

370513 

370698 

370883 

186 

6 

371068 

371253 

371437 

371622371806 

1991 

2175 

2360 

2544 

2728 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

380030 

181 

240 

380211 

380392 

380573 

380754 

380934 

381115 

331296 

381476 

381656 

381837 

181 

1 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

2 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

6249 

5428 

179 

3 

5606 

5785 

5964 

6142|  6321 

6499 

6677 

6856 

7034 

7212 

178 

4 

7390 

7568 

7746 

7923:  810 

8279 

8456 

8634 

8811 

8989 

178 

5 

9166 

9343 

9520 

9698   9S75 

390051 

390223 

390405 

390582 

390759 

177 

6 

390935 

391112 

391288 

3914641391641 

1817 

1993 

2169 

23451  2521 

176 

7 

2697 

2873 

3048 

3221 

3400 

3575 

3751 

3926 

4101 

4277 

176 

8 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

176 

9 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

397940 

398114 

398287 

398461 

398634 

398808 

393981 

399154 

399328 

399501 

173 

1 

9674 

9847 

400020 

400192400365 

400538 

400711 

400883 

401056 

401228 

173 

2 

401401 

401573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

3 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

171 

4 

483-1 

5005 

5176 

5346 

5517 

5688   5858 

6029 

6199 

6370 

171 

6 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

6 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426   9595 

9764 

169 

7 

9933 

410102 

410271 

410440 

410609 

410777 

410946 

411114 

411283 

411451 

169 

8 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

9 

3300 

3467 

3635 

3303 

3970 

4137 

430f 

4472 

4639 

4806 

167 

260 

414973 

415140 

415307 

415474 

415641 

415808 

415974 

416141 

416308 

416474 

167 

1 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

2 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

3 

9956 

420121 

420286 

420451 

420616 

420781 

420945 

421110 

421275 

421439 

165 

4 

421604 

1768 

1933 

2097 

2261 

2426 

2590 

2764 

2918 

3082 

164 

5 

3246  3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

e 

4882 

5045   5108 

5371 

5534 

5697 

5860 

6023 

6186 

6349 

163 

6511 

6674 

6836   6999 

7161 

7324 

7486 

7648 

7811 

7973 

162 

8 

8135 

8297 

8459 

8621 

8783 

8944   9106 

9268 

9429 

9591 

162' 

9 

9752 

9914 

430075 

430236 

430398 

430559 

43072C 

430881 

431042 

431203 

161 

WO 

431364 
2969 

431525 
3130 

431685 
3290 

431846 
3450 

432007 
3610 

432167 
3770 

432323 
393U 

432488 

4090 

432649 
4249 

432809 
4409 

161 
160 

( 

4569 

4729 

4888 

5018   5207 

5367   5526 

5685 

5844 

6004 

159 

j 

6163 

6322 

6481 

6640   6799 

6957 

7116 

7275 

7433 

7592 

159 

4 

7751 

7909 

8067 

8226  1  8334 

3542 

870  li  8859,  9017 

9175 

158 

5 
6 

9333 

440909 

9491 
441066 

9648 
441224 

9806   9964 
441381  1441533 

4401  22  [  440279 
1695   1852 

1  440437  440594 
2009:  2166 

440752 
2323 

158 
157 

• 

2480 

2637 

2793 

2950   3106 

3263   3419 

3576;  3732 

3889 

157 

6 

4045 

4201 

4357 

4513   46691  4825  i  4981 

51371  5293 

6449 

156 

5604 

5760 

5915 

6071 

6226 

6382 

-  6537 

6692 

6848 

7003  155 

Ho 

0 

1  1  2 

3 

* 

5 

6 

7 

8 

9  jDitf. 

TABLE   XIY.       LOGARITHMS    OF   NtTMBERS. 


225 


Mo 

O 

1 

8 

3 

4 

5 

6 

7 

8 

9 

Diflf. 

280 

447168 

447313 

447468 

447623 

447778 

447933 

448088 

448242 

448397 

448552 

155 

1 

8706 

8861 

90151  9170 

9324 

9478 

9633 

9787 

9941 

450095 

154 

2 

450249 

450403 

450557450711 

450865 

451018 

451172 

451326 

451479 

1633 

154 

3 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

6 

4845 

4997 

6160 

6302 

6454 

5606 

6768 

6910 

6062 

6214 

152 

6 

6366 

6518 

6670 

6821 

6973 

7125 

7276 

7428 

7679 

7731 

152 

7 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

8 
9 

9392 
460898 

9543 

461048 

9694 
461198 

9845 
461348 

9995 
461499 

460146 
1649 

460296 
1799 

460447 
1948 

460697 
2098 

460748 
2248 

151 
150 

290 

462398 

462648 

462697 

462847 

462997 

463146 

463296 

463445 

463594 

463744 

150 

1 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

6085 

5234 

149 

2 

6383 

5532 

6680 

6829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

7312 

7460 

7608 

7766 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

5 

9822 

9969 

470116 

470263 

470410 

470557 

470704 

470851 

470998 

471145 

147 

6 

471292 

471438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

4663 

4799 

4944 

5090 

5235 

5381 

6526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

146 

300 

477121 

477266 

477411 

477566 

477700 

477844 

477989 

478133 

478278 

478422 

145 

1 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

2 

480007 

480151 

480294 

480438 

480582 

480725 

480869 

481012 

481156 

481299 

144 

3 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

4 

2874 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157 

143 

5 

4300 

4442 

4585 

4727 

4869 

6011 

6163 

6295 

6437 

6579 

142 

6 

5721 

6863 

6005 

6147 

6289 

6430 

6672 

6714 

6856 

6997 

142 

7 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

141 

8 

8651 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

9 

9963 

490099 

490239 

490330 

490620 

490661 

490801 

490941 

491081 

491222 

140 

310 

491362 

491602 

491642 

491782 

491922 

492062 

492201 

492341 

492481 

492621 

140 

1 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

2 

4155 

4294 

4433 

4572 

4711 

4850 

4989 

6128 

6267 

6406 

139 

3 

5544 

6683 

6322 

6960 

6099 

6233 

6376 

6515 

6653 

6791 

139 

4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

5 

8311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9660 

138 

6 

9687 

9824 

9962 

500099 

500236 

500374 

600511 

500648 

500785 

600922 

137 

7 

601059 

601196 

501333 

1470 

1607 

1744 

1880 

2017 

2164 

2291 

137 

8 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3655 

136 

9 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4678 

6014 

136 

320 

505150 

505286 

505421 

605557 

605693 

505828 

605964 

606099 

506234 

606370 

136 

1 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

2 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8799 

8934 

9068 

135 

3 

9203 

9337 

9471 

9606 

9740 

9874 

510009 

510143 

510277 

610411 

134 

4 

610545 

510679 

510313 

510947 

511081 

511215 

1349 

1482 

1616 

1750 

134 

6 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

133 

6 

3213 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4415 

133 

7 

4548 

4631 

4313 

4946 

5079 

5211 

5344 

6476 

5609 

5741 

133 

8 

5874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

132 

9 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

8261 

8382 

132 

330 

818614 

518646 

518777 

518909 

519040 

519171 

519303 

519434 

519566 

519697 

131 

1 

9823 

9959 

520090 

520221 

520353 

520484 

520615 

520745 

520876 

621007 

131 

2 

621138 

521269 

1400 

1530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

3 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

4 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

6 

5045 

6174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

6 

6339 

6469 

6598 

6727 

6356 

6935 

7114 

7243 

7372 

7501 

129 

7 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8531 

8660 

8788 

129 

8 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

530072 

123 

9 

530200 

530323 

530456 

530584 

530712 

530840 

530968 

531096 

531223 

1351 

128 

No 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

IKff. 

226 


TABLE   XIV.       LOGARITHMS    OF   NUMBERS. 


No. 

O 

1 

2 

9 

Dlff. 

340 

531479 

531607 

31734 

531862  531990 

532117 

532245 

)32372 

632500 

532627 

128 

1 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

2 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

6167 

127 

3 

5294 

5421 

5547 

5674 

5800 

6927 

6053 

6180 

6306 

6432 

126 

4 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

6 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

6 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

5400791540204 

125 

7 

540329 

540455 

540580 

£40705 

540830 

540955 

541080 

641205 

1330  1454 

126 

3 

1679 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

9 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

390 

644068 

544192 

644316 

544440 

544564 

544688 

544812 

644936 

645060 

645183 

124 

1 

6307 

5431 

5555 

5678 

5802 

6925 

6049 

6172 

6296 

6419 

124 

2 

6643 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

3 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

S635 

8758 

8881 

123 

4 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

660106 

123 

5 

550228 

650351 

550473 

650595 

650717 

650840 

650962 

551084 

651206 

1328 

122 

6 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

7 

2663 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

8 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

9 

6094 

5215 

5336 

5457 

6578 

6699 

5820 

6940 

6061 

6182 

121 

360 

666303 

666423 

656544 

556664 

666785 

666905 

657026 

667146 

657267 

657387 

120 

1 

7607 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

2 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

b 

9907 

660026 

560146 

560265 

660385 

560504 

660624 

660743 

660863 

660982 

119 

4 

661101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

5 

8293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

6 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

7 

4666 

4784 

4903 

6021 

6139 

6257 

6376 

6494 

6612 

6730 

118 

8 

6848 

6966 

6034 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

9 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

870 

668202 

668319 

668436 

668554 

568671 

668788 

668905 

669023 

669140 

669257 

117 

1 

9374 

9491 

9608 

9725 

9842 

9959 

670076 

670193 

670309 

670426 

117 

2 

670543 

670660 

570776 

670893 

671010 

571126 

1243 

1359 

1476 

1592 

117 

3 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2765 

11(5 

4 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

380u 

3915 

116 

6 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4967 

6072 

116 

6 

6188 

6303 

6419 

6534 

5650 

6765  6880 

6996 

6111 

6226 

115 

7 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

8 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

115 

9 

6639 

8764 

8868 

8983 

9097 

9212 

9326 

9441 

9565 

9669 

114 

380 

679784 

679898 

680012 

580126 

680241 

680355 

680469 

580583 

680697 

580811 

114 

1 

680925 

581039 

1153 

1267 

1381 

1495 

1608 

722 

,  1836 

1950 

114 

2 

2063 

2177 

2291 

2404 

2518 

2631 

2746 

J58 

#972  3085 

114 

3 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

J992 

4105  4218 

4 

4331 

4444 

4557 

4670 

4783 

4896 

6009 

6122 

6235  5348 

12 

6 

6461 

6574 

5686 

6799 

5912 

6024 

6137 

6250 

6362 

6475 

13 

6 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

.12 

7 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112 

8 

8832 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

112 

9 

9950 

690061 

590173 

690284 

590396 

690507 

590619 

690730 

690842 

690953 

112 

390 

691065 

F91176 

691287 

591399 

661510 

691621 

691732 

691843 

691955 

692066 

111 

1 

217? 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

« 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

111 

J 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

6386 

110 

4 

6496 

6606 

6717 

5827:  6937 

6047 

6157 

6267 

6377 

6487 

110 

5 

6597 

6707 

6817 

6927   7037 

7146 

7256 

7366 

7476 

7586 

110 

( 

7695 

7805 

7914 

8024   8134 

8243 

8353 

8462 

8572 

8681 

110 

j 

8791 

8900 

9009 

9119  9228 

9337 

9446 

9556 

9665 

9774 

109 

8 

9883 

9992 

60010 

600210  600319 

60042S 

600537 

600646 

600755 

600864 

109 

9 

600973 

601082 

1191 

1299   1408 

1517 

1625 

1734 

1843 

1951 

109 

Mo 

0 

1 

3 

3  !  4 

5 

6 

7 

8  |  9 

Diff. 

TABLE    XIV.       LOGARITHMS    OF    NUMBERS. 


221 


|No. 

0 

1 

3    3  !  4 

5 

6 

7    8 

9   Dlff 

400 

602060  602169 

602277.602336602494 

602603 

602711 

602819  602928 

603036 

108 

1 

3144   3253 

3361  ;  3469  3577 

3636 

3794 

3902.  4010 

4118 

108 

2  4226 

4334 

4442  4550  4658 

4766 

4874 

4982i  5039 

5197 

108 

3  5305 

5413 

5521  i  5628 

5736 

5844 

5951 

6059 

6166 

6274 

108 

4  6381 

6489 

6596:  6704 

6811 

6919 

7026 

7133 

7241 

7348 

107 

51  7455   7562 

7669   7777 

7884 

7991 

8098 

8205 

8312 

8419 

107 

8  8526  8633 

8740  !  8847 

8954 

9061 

9167 

9274  9381 

9488 

107 

7i  9594 

9701 

9808  9914 

610021 

610128 

610234 

610341  610447 

610554 

107 

8610660 

610767 

610373  610979 

1086 

1192 

1298 

1405 

1511 

1617 

106 

9 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

612784 

612890 

612996  613102 

613207 

613313 

613419 

613525 

613630 

613736 

106 

1 

3842 

3947 

4053  4159 

4264 

4370 

4475 

4581 

4686 

4792 

106 

9 

4897 

5003 

51081  5213 

5319 

5424 

5529 

5634 

5740 

5845 

105 

3 

5950 

6055 

6160  6265 

6370 

6476 

6581 

6686 

6790 

6895 

105 

4 

7000 

7105 

7210  7315 

7420 

7525 

7629 

7734 

7839 

7943 

105 

5 

8048 

8153 

8257  8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

6 

9093 

9198 

9302|  9406  9511 

9615 

9719 

9824 

9928 

620032 

104 

7 

620136 

620240 

620344  620443  620552 

620656 

620760 

620364 

620963 

1072 

104 

8 

1176 

1280 

1334   14-38   1592 

1695 

1799 

1903 

2007 

2110 

104 

9 

2214 

2318 

242  lj  25251  2628 

2732 

2835 

2939 

3042 

3146 

104 

420 

623249 

623353 

623456 

623559 

623663 

623766 

623869 

623973 

624076 

624179 

103 

1 

4282 

4385 

4483 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

103 

3 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7263 

103 

4 

7366 

7463 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

102 

5 

8339 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

6 

9410 

9512 

9613 

9715 

9817 

9919 

630021 

630123 

630224 

630326 

102 

7 

630423 

630530 

630631 

630733 

630835 

630936 

1038 

1139 

1241 

1342 

102 

8 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

101 

9 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

101 

430 

633468 

633569 

633670 

633771 

633872 

633973 

634074 

634175 

634276 

634376 

101 

I 

4477 

4573 

4679 

4779 

4830 

4931 

5081 

5182 

5283 

5383 

101 

2 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

100 

3 

6438 

6533 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

100 

4 

7490 

7590 

7690!  7790 

7890 

7990 

8090 

8190 

8290 

8389 

100 

5 

8489 

8539 

8689  S  8789 

8888 

8988 

9088 

9188 

9287 

9387 

100 

6 

9436 

9586 

9686  9785  9885 

9984 

640084 

640183 

640283 

640382 

99 

7 

640481 

640531 

640680  640779  640879 

640978 

1077 

1177 

1276 

1375 

99 

8 

1474 

1573 

1672   1771 

1871 

1970 

2069 

2168 

2267 

2366 

99 

9 

2465 

2563 

2662  2761 

2360 

2959 

3058 

3156 

3255 

3354 

99 

440 

643453 

643551 

643650 

643749 

643847 

643946 

644044 

644143 

644242 

644340 

98 

1  4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

5324 

98 

2 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306, 

93 

3 

6404 

6502 

6600  6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

4 

7333 

7481 

7579!  7676 

7774 

7872 

7969 

8067 

8165 

8262 

98 

5 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

97 

6 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

650016 

650113 

650210 

97 

7 

650308 

650405 

650502  650599 

650696 

650793 

650890 

0987 

1084 

1181 

97 

8 

1278 

1375 

1472   1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

9 

2246 

2343 

2440  2536 

2633 

2730 

2826 

2923 

3019 

3116 

97 

460 

653213 

653309 

653405 

653502 

653598 

653695 

653791 

653888 

653984 

654080 

96 

1 

4177 

4273 

4369  4465  4562 

4658 

4754 

4850 

4946 

5042 

96 

2 

5138 

5235   5331   5427   5523 

5619 

5715 

5810 

5906 

6002 

96 

3 

6093 

6194!  6290  6386  6482 

6577 

6673 

6769 

6864 

6960 

96 

4 

7056 

7152   7247   7343   7438 

7534 

7629 

7725 

7820 

7916 

96 

5 

8011   8107   8202   8293  8393 

8488 

8584 

8679 

8774 

8870 

95 

6 

8965  9060   9155  9250!  9346 

9441 

9536 

9631 

9726 

9821 

95 

7 

99  16  66001  11660  106  660201  660296 

660391 

660486 

660581 

660676 

660771 

95 

8660865  0960|  1055J  1150   1245   1339 

1434 

1529 

1623 

1718 

95 

9   1813;  1907 

2002 

2096 

2191  1  2286 

2380 

2475 

2569 

2663 

96  1 

a  0.1  o  j  i 

3 

3 

4    5 

6 

T 

8 

9 

DUT  1 

228 


TABLE   XIV.       LOGARITHMS   OP   NUMBERS. 


No. 

O 

i  |  a 

3 

4 

5  1 

6 

r 

8  1  9 

Diff. 

460 

662758 

662852  662947 

663041 

663135 

663230 

663324 

663418 

663512  663607 

94 

1 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

94 

2 

4642 

4736 

4830 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

3 

5581 

5675 

5769 

5862 

5956 

6050 

6143 

6237 

6331 

6424 

94 

4 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

94 

5 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

93 

6  8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

93 

7l 
8 

9317 

670246 

9410 
670339 

9503 
670431 

9f96 
670524 

9689 
670617 

9782 
670710 

9875 
670802 

9967 
670895 

670060 
0988 

670153 
1080 

93 
93 

9 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

672190 

672283 

672375 

672467 

672560 

672652 

672744 

672836 

672929 

92 

1 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

92 

2 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

92 

3 

4861 

4953 

5045 

5137 

6228 

5320 

5412 

5503 

5595 

5687 

92 

4 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

92 

5 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7616 

91 

6 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

91 

7 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9155 

9246 

9337 

91 

8 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

680063 

680154 

680245 

91 

9 

680336 

680426 

680517 

680607 

680698 

680789 

680879 

0970 

1060 

1151 

91 

480 

681241 

681332 

681422 

681513 

681603 

681693 

681784 

681874 

681964 

682055 

90 

1 

2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

90 

2 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

3 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

4 

4845 

4935 

6025 

5114 

6204 

5294 

5383 

5473 

6563 

6652 

90 

5 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

6 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

89 

7 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

89 

8 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

89 

9 

9309 

9398 

9486 

9576 

9664 

9753 

9841 

9930 

690019 

690107 

89 

490 

690196 

690285 

690373 

690462 

690550 

690639 

690728 

690816 

690905 

690993 

89 

1 

1081 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

88 

2 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

88 

3 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

88 

4 

3727 

3315 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

88 

5 

4605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

88 

6 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

7 

6356 

6444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

87 

8 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

9 

8101 

8188 

8275 

8362 

8449 

8536 

8622 

8709 

8796 

8883 

87 

500 

698970 

699057 

699144 

699231 

699317 

699404 

699491 

699578 

699664 

699751 

87 

1 

9838 

9924 

700011 

700098700184 

700271 

700358 

700444 

700531 

700617 

87 

2 

700704 

700790 

0877 

09631  1050 

1130 

1222 

1309 

1395 

1482 

86 

3 

1568 

1654 

i741 

1827!  1913 

1999 

2086 

2172 

2258 

2344 

86 

4 

2431 

2517 

2603 

2689!  2775 

2861 

2947 

3033  3119 

3205 

86 

5 

3291 

3377 

3463 

3549!  3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4408!  4494 

4579 

4665 

4751 

4837 

4922 

86 

7 

5008 

5094 

6179 

5265  6350 

5436 

6522 

5607 

5693 

5778 

86 

8 

5864 

5949 

6035 

6120  6206 

6291 

6376 

6462 

6547 

6632 

85 

9 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7316 

7400 

7485 

85 

510 

707570 

707655 

707740 

707826 

707911 

707996 

708081 

708166 

708251 

708336 

85 

1 

8421 

8506 

8591 

8676  8761 

8846 

8931 

9015 

9100 

9185 

85 

2 

9270 

9355 

9440 

9524  9609 

9694 

9779 

9863 

9948 

710033 

85 

3 

710117 

710202 

710287 

710371  710456 

710540 

710625 

710710  710794 

0879 

85 

4 

0963 

1048 

1132 

1217   1301 

1385 

1470 

1554 

1639 

1723 

84 

5 

1807 

1892 

1976 

2060  2144 

2229 

2313 

2397 

2481 

2566 

84 

6 

2650 

2734 

2818 

2902  2986 

3070 

3154 

3238 

3323 

3407 

84 

7 

3491 

3575 

3659 

3742  3826 

3910 

3994 

4078 

4162 

4246 

84 

8 

4330 

4414 

4497 

4581   4665 

4749 

4833 

4916 

6000 

6084 

84 

9 

5167 

5251 

5335 

5418  5502 

5586 

5669 

5753 

5836 

5920 

84 

Ko 

0 

1 

3 

3    4 

5 

6 

7 

8 

e 

JHff. 

TABLE   XIV.       LOGARITHMS   OF   NUMBERS. 


229 


520 

716003716087 

716170  716254 

716337 

716421 

716504 

716588 

8 
716671 

9 

716754 

DHL 

83 

1 

6838  6921 

7004   7088 

7171 

7254 

7338 

7421 

7504 

7587 

82 

2 

7671   7754 

7837;  7920 

8003 

8086 

8169 

8253 

8336 

8419 

82 

3 

8502 

8585 

8668J  8751 

8834 

8917 

9000 

9083 

9165 

9248 

82 

4 
5 

9331 
720159 

9414 
720242 

9497   9580 
720325720407 

9663 
720490 

9745 
720573 

9828 
720655 

9911 
720738 

9994 
720821 

720077 
0903 

83 
83 

6 

7 

0986 
1811 

1068 
1893 

1151J  1233 
1975  2058 

1316 
2140 

1398 
2222 

1481 
2305 

1563 
2387 

1646 
2469 

1728 
2552 

82 
82 

8 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

9 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

724276 

724358 

724440 

724522 

724604 

724685 

724767 

724849 

724931 

725013 

82 

1 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

82 

2 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

82 

3 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

6 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

6 

9165 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

981  J 

9892 

81 

7 

9974 

730055 

730136 

730217 

730298 

730378 

730459 

730540 

730621 

730702 

81 

8 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

9 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

81 

640 

732394 
3197 

732474 
3278 

732555 
3358 

732635 
3438 

732715 
3518 

732796 
3593 

732876 

3679 

732966 
3759 

733037 
3839 

733117 
3919 

80 
80 

2 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4660 

4640 

4720 

80 

3 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

6359 

5439 

5519 

80 

4 

5599 

6679 

6759 

5838 

5918 

6998 

6078 

6157 

6237 

6317 

80 

6 

6397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7113 

80 

6 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7906 

79 

7 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

79 

8 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

9 

9572 

9651 

9731 

9810 

9389 

9963 

740047 

740126 

740205 

740284 

79 

660 

1 

740363 
1152 

740442 
1230 

740521 
1309 

740600 
1388 

740678 
1467 

740757 
1546 

740836 
1624 

740915 
1703 

740994 
1782 

741073 
1860 

79 
79 

2 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

79 

3 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

4 

3510 

35.SS 

3667 

3745 

3823 

3902 

3930 

4058 

4136 

4216 

78 

6 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

6 

5075 

6153 

5231 

5309 

5387 

6465 

5543 

5621 

5699 

5777 

78 

7 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

8 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7179 

7256 

7334 

78 

9 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

78 

560  748188 

748266 

748343 

748421 

748498 

748576 

748653 

748731 

748808 

748885 

77 

1 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

77 

2 
3 

9736 

750508 

9814 
750586 

9891 
750663 

9968 
750740 

750045 
0817 

750123 
0894 

750200 
0971 

750277 
1048 

750354 
1125 

750431 
1202 

77 
77 

4 

1279 

1356 

1433 

1510 

1587 

1664 

1741 

1818 

1895 

1972 

77 

6 

2048 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

77 

6 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

77 

7 

3583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

77 

8 

434  S 

4425 

4501 

4578 

4654 

4730 

4807 

4883 

4960 

6036 

76 

9 

5112 

5189 

5265 

5341 

6417 

6494 

5570 

5646 

5722 

6799 

76 

570 

755875 

755951 

756027 

756103 

756180 

756256 

756332 

756408 

756484 

756560 

76 

1 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

76 

2 

7396 

7472 

7548!  7624 

7700 

7775 

7851 

7927 

8003 

8079 

76 

3 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

76 

4 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

76 

5 

; 

9668 
760422 

9743 
760498 

9819 
760573 

9894 
760649 

9970 
760724 

760045 
0799 

760121 
0375 

760196 
0950 

760272 
1025 

760347 
1101 

76 
76 

7   1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

76 

8  1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

76 

9  2679 

2754 

2829 

2904 

2978 

3053 

3128 

3203 

3278 

3353 

76 

No.)  0 

1 

3  1  3 

4 

5 

6 

7 

8 

9 

Dlff. 

•230 


TABLE    XIV.       LOGARITHMS   OF   NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dltt'.  j 

1680 

763428 

"63503 

763578 

763653  763727 

763802 

763877 

763952 

764027 

764101 

76 

1 

4176 

4251 

4326 

4400  4475 

4550 

4624 

4699 

4774 

4848 

76 

2 

4923 

4998 

5072 

5147 

6221 

6296 

6370 

5445 

5520 

5594 

76 

3 

5669 

5743 

6818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

74 

4 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

74 

5 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

74 

6 

7895 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490|  8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9G08 

9082 

9156 

9230 

9303 

74 

8 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

770042 

74 

9 

770115 

770189 

770263 

770336 

770410 

770484 

770557 

770631 

770705 

0778 

74 

690 

770852 

770926 

770999 

771073 

771146 

771220 

771293 

771367 

771440 

771514 

74 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

3 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

6 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

6028 

5100 

6173 

73 

6 

6246 

6319 

5392 

5465 

5538 

5610 

6683 

6756 

6829 

6902 

73 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

73 

6 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

9 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

72 

600 

778151 

778224 

778296 

778368 

778441 

778513 

778585 

778658 

778730 

778802 

72 

1 

8874 

8947 

9C19 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

2 
3 

9596 
780317 

9669 

780389 

9741 
780461 

9813 
780533 

9885 
780605 

9957 

780677 

780029 
0749 

780101 
0821 

780173 
0893 

780245 
0965 

72 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

72 

6 

1756 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

6 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

71 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

6116 

6187 

6259 

71 

610 

786330 

785401 

785472 

785543 

785615 

785686 

786757 

785828 

786899 

785970 

71 

1 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6638 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

71 

3 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

71 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

71 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

71 

6 

9581 

9651 

9722 

9792 

9863 

9933 

790004 

790074 

790144 

790215 

70 

7 

790285 

790356 

790426 

790496 

790567 

790637 

0707 

0778 

0848 

0918 

70 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

70 

9 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

792462 

792532 

792602 

792672 

792742 

792812 

792882 

792952 

793022 

70 

1 

3092 

3162 

3231 

330r:  3371 

3441 

3511 

3581 

3651 

3721 

70 

2 

3790 

3860 

3930 

4000  4070 

4139 

4209 

4279 

4349 

4418 

70 

3 

4488 

4558 

4627 

4697  4767 

4836 

4906 

4976 

6045 

5115 

70 

4 

5185 

5254 

5324 

5393  5463 

5532 

5602 

5672 

5741 

5811 

70 

5 

5880 

5949 

6019 

6088  6158 

6227 

6297 

6366 

6436 

6505 

69 

6 

6574 

6644 

6713 

6782  6852 

6921 

6990 

7060 

7129 

7198 

69 

7 

7268 

7337 

7406 

7475  7545 

7614 

7683 

7752 

7821 

7890 

69 

8 

7960 

8029 

8098 

8167  8236 

8305 

8374 

8443 

8513 

8582 

69 

9 

8651 

8720 

8789 

8858  8927 

8996 

9065 

9134 

9203 

9272 

69 

630 

799341 

799409 

799478 

799547  799616 

799685 

799754 

799823 

799892 

799961 

69 

1 

800029 

800098 

800167 

800236  800305 

800373 

800442 

800511 

800580 

800648 

69 

c 

0717 

0786 

0854 

0923  0992 

1061 

1129 

1198 

1266 

1335 

69 

: 

1404 

1472 

1541 

1609   1678 

1747 

1815 

1884 

1952 

2021 

69 

* 

2089 

2158 

2226 

2295  2363 

2432 

2500 

2568 

2637 

2706 

68 

5 

2774 

2842 

2910 

2979   3047 

3116 

3184 

3252 

3321 

3389 

68 

6 

3457 

3525 

3594 

3662  3730 

3798 

3867 

3935 

4003 

4071 

68 

' 

4139 

4208 

4276 

4344  4412 

4480 

4548 

4616 

4685 

4753 

68 

8 

4821 

4889  |  4957 

5025  5093 

5161 

5229 

5297 

5365 

6433 

68 

9 

6501 

5569  |  5637 

5705   5773 

5841 

5908 

6976 

6044 

6112 

68 

j  No 

O 

1   1  * 

3    4 

5 

6 

7 

8 

9 

DMT. 

TABLE   XIV.       LOGARITHMS   OF   NUMBERS. 


231 


Ho.  0  | 

1 

a 

8 

9 

Diff. 

640 

306180 

806248 

306316 

306334 

06451 

06519 

06587 

06655 

06723 

806790 

68 

6858 

69^6 

6994 

7061 

7129 

7197 

7264 

7332 

7400 

7467 

68 

2 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

68 

3 

8211 

8279 

8346 

8414 

8481 

8549 

861G 

8684 

8751 

8818 

67 

4 

8886 

8953 

9021 

9088 

9156 

9223 

9290 

9358 

9425 

9492 

67 

5 

9560 

9827 

9694 

9762 

9829 

9896 

9964 

810031 

10098 

810165 

67 

6 

810233 

810300 

810367 

810434 

310501 

810569 

10636 

0703 

0770 

0837 

67 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1576  1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

67 

650 

812913 

812980 

813047 

813114 

813181 

813247 

813314 

813381 

813448 

813514 

67 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

67 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

6378 

5445 

5511 

66 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

66 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

66 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

7 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

66 

8 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

66 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

819544 

819610 

819676 

819741 

819807 

819873 

319939 

820004 

820070 

820136 

66 

1 

820201 

820267 

820333 

820399 

820464 

820530 

820595 

0661 

0727 

0792 

66 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

66 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

65 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

65 

6 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

65 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

65 

7 

4126 

4191 

4256 

4321 

4336 

4451 

4516 

4581 

4646 

4711 

65 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

65 

1 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5830 

5945 

6010 

65 

670 

826075 

826140 

826204 

826269 

826334 

826399 

826464 

826528 

826593 

826658 

65 

1 

6723 

6787 

6852 

6917 

6931 

7046 

7111 

7175 

7240 

7305 

65 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

65 

2 

8015 

8080 

8144 

8209 

8273 

8333 

8402 

8467 

8531 

8595 

64 

4 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

5 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

64 

e 

9947 

830011 

830075 

830139 

830204 

830268 

830332 

830396 

830460 

830525 

64 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

8 

1230 

1294 

1353 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

832509 

832573 

832637 

832700 

832764 

832828 

832892 

832956 

833020 

833083 

64 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

64 

2 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

3 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

4 

5056 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

5564 

5627 

63 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

6 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6394 

63 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

690 

838849 

838912 

833975 

839038 

839101 

839164 

839227 

839289 

839352 

839415 

63 

I 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

840043 

63 

2 

840106 

840169 

840232:840294 

840357 

840420 

840482 

840545 

840608 

0671 

63 

3 

0733 

0796 

0859!  0921 

0984 

1046 

1109 

1172 

1234 

1297 

63 

4 

1359 

1422 

1485J  1547 

1610 

1672 

1735 

1797 

1860 

1922 

63 

5 

1935 

2047 

2110;  2172 

2235 

229 

2360 

2422 

2484 

2547 

62 

6 

2609 

267:. 

27341  2796 

2859 

292 

2983 

3046 

3108 

3170 

62 

7 

3232 

329o 

3357 

3420 

3482 

354 

3606 

3669 

373 

3793 

62 

8 

3855 

39  IS 

398C 

4042 

4104 

416 

4229 

429 

4353 

4415 

62 

9  4477 

453 

4601 

4664 

472 

478 

4850 

4912 

4974 

5036 

62 

No.]  0 

1 

3 

3 

* 

5 

6 

7 

8 

e 

Dtff. 

TABLE   XIV.       LOGARITHMS    OF   NUMBERS. 


No 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

700 

845098 

845160 

845222 

845284 

845346 

845408 

84547C 

845532 

845594 

845656 

62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

609C 

6151 

6213!  6275 

62 

2 

6337 

639D  6461 

6523 

6585 

6646 

6708 

6770 

6832   6894 

62 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

62 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8l2fe 

62 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

62 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

61  I 

7 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61  ' 

8 

850033 

850095 

850156 

850217 

850279 

850340 

850401 

850462 

850524 

850585 

61 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

851258 

851320 

851381 

851442 

851503 

851564 

851625 

851686 

851747 

851809 

61 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

61 

4 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

61 

5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4862 

61 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

7 

5519 

5580 

5640 

5701 

5761 

5822 

6882 

5943 

6003 

6064 

61 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

857393 

857453 

857513 

857574 

857634 

857694 

857755 

857815 

857875 

60 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

3 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

860038 

860098 

860158 

860218 

860278 

60 

6 

860338 

860398 

860458 

860518 

860578 

0637 

0697 

0757 

0817 

0877 

60  i 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1476 

60 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

60  1 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

60 

730 

863323 

863382 

863442 

863501 

863561 

863620 

863680 

863739 

863799 

863858 

69 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

6045 

69 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

6519 

5578 

5637 

59 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

69 

5 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

69 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

69 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

69 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

69 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9066 

9114 

9173 

69 

740 

869232 

869290 

869349 

869408 

869466 

869525 

869584 

869642 

869701 

869760 

69 

1 

9818 

9877 

9935 

9994 

870053 

870111 

870170 

870228 

870287 

870345 

69  1 

2 

870404 

870462 

870521 

870579 

0638 

0696 

0755 

0813 

0872 

0930 

68 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

68 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

6 

2739 

2797 

2855 

2913 

2972 

3030 

308S 

3146 

3204 

3262 

58 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

63 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

68 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

6003 

68 

760 

875061 

875119 

875177 

875235 

875293 

875351 

875409 

875466 

875524 

875582 

ts 

1 

5640 

5698  5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

68 

2 

6218 

6276  6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

68 

3 

6795 

6853  1  6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

68 

4 

7371 

7429;  7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

68 

5 

7947 

8004'  8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

67 

6 

8522 

8579  8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

7 

9096 

9153   9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

67 

8 
9 

9669 
880242 

9726   9784 
880299  880356 

9841 
880413 

9898 
8S0471 

9956 

880528 

880013 
0585 

880070 
0642 

880127 
0699 

880185 
0756 

67 
67 

No. 

O 

1    2 

3 

4 

5 

6 

7 

8 

9 

Diff.  1 

TABLE   XIV.       LOGARITHMS   OF   NUMBERS. 


233 


Mo. 

0 

1 

8 

3 

4 

5 

6 

y 

8 

9 

Dlff. 

760 

880814 

830871 

830928 

880985 

881042 

881099 

881156 

881213 

881271 

881328 

67 

1 

1335 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

57 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

67 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866  2923 

2980 

3037 

67 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434  3491 

3548 

3605 

67 

5 

3661 

3718 

3775 

3832 

3888 

3945 

4002  4059 

4115 

4172 

57 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569  4625 

4682 

4739 

57 

7 

4795 

4852 

4909 

4965 

5022 

5078 

5135J  5192 

5248 

5305 

57 

i  8 

5361 

5418 

5474 

5531 

5587 

6644 

6700 

5757 

5813 

5870 

57 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

66 

770 

886491 

886547 

886604 

886660 

886716 

886773 

886829 

886385 

886942 

886998 

66 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

66 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

66 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

66 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

66 

5 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

66 

6 

9862 

9918 

9974 

890030 

890086 

890141 

890197 

890253 

890309 

890365 

66 

7 

890421 

890477 

890533 

0589 

0645 

0700 

0756 

0812 

0863 

0924 

66 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

66 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

66 

780 

892095 

892150 

892206 

892262 

892317 

892373 

892429 

892484 

892540 

892596 

66 

1 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

66 

2 

3207 

3262 

3318 

3373 

3429 

3434 

3540 

3595 

3651 

3706 

66 

3 

3762 

3817 

3873 

3928 

3934 

4039 

4094 

4160 

4205 

4261 

65 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

65 

5 

4870 

4925 

4980 

6036 

6091 

5146 

5201 

6257 

5312 

6367 

66 

6 

6423 

6478 

5533 

5588 

5644 

6699 

6764 

6809 

5864 

5920 

66 

7 

6975 

6030 

6085 

6140 

6195 

6261 

6306 

6361 

6416 

6471 

66 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6867 

6912 

6967 

7022 

66 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7617 

7672 

66 

790 

897627 

897682 

897737 

897792 

897847 

897902 

897967 

898012 

898067 

898122 

66 

1 

8176 

8231 

8286 

8341 

8396 

S451 

8606 

8561 

8615 

8670 

65 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9064 

9109 

9164 

9218 

66 

3 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

66 

4 

9821 

9875 

9930 

9985 

900039 

900094 

900149 

900203 

900258 

900312 

66 

6 

900367 

900422 

900476 

900531 

0586 

0640 

0695 

0749 

0804 

0859 

66 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

66 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

64 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

64 

9 

9547 

2601 

2655 

2710 

2764 

2318 

2873 

2927 

2981 

3036 

64 

800 

903090 

903144 

903199 

903253 

903307 

903361 

903416 

903470 

903524 

903578 

64 

1 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

64 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

64 

3 

4716 

4770 

4824 

4878 

4932 

4936 

5040 

5094 

5148 

5202 

64 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

64 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

64 

6 

6335 

6389 

6443 

6497 

6551 

6604 

6653 

6712 

6766 

6820 

64 

7 

6874 

69£7 

6931 

7035 

7039 

7143 

7196 

7250 

7304 

7358 

54 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

64 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

3431 

64 

610 

908485 

908539 

908592 

90S646 

903699 

908753 

908807 

908860 

908914 

908967 

64 

1 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

64 

2 

9556 

9610 

9663 

9716 

9770 

9323 

9877 

9930 

9984 

910037 

63 

3 

910091 

910144 

910197 

910251 

910304 

910358 

910411 

910464 

910518 

0571 

63 

4 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

63 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

53 

6 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

63 

7 

2222 

2275 

2328 

2331 

2435 

2438  2541 

2594 

2647 

2700 

63 

8 

2753 

2806 

2859 

2913 

2966 

3019!  3072 

3125 

3178 

3231 

63 

9 

3234 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

63 

Nat  O 

1 

3 

3 

4 

5 

o 

7 

8 

9 

Dttf. 

TABLE   XIV.       LOGARITHMS   OF   NUMBERS. 


Mo. 

O 

1 

£ 

3 

4 

5 

6 

r 

8 

9  jDiff. 

820 

913814 

913867 

913920 

913973 

914026 

914079 

914132 

914184 

914237 

914290 

63 

1 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

53 

2 

4872 

4925 

4977 

5030 

6083 

5136 

5189 

5241 

5294 

5347 

63 

3 

5400 

5453 

6505 

5558 

5611 

5664 

6716 

6769 

5822 

5875 

63 

4 

5927 

6980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

5 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

6 

6980 

7033 

7085 

7133 

7190 

7243 

7295 

7348 

7400 

7453 

53 

7 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

62 

8 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

52 

9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

919078 

919130 

919183 

919235 

919287 

919340 

919392 

919444 

919496 

919549 

62 

1 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

920019 

920071 

52 

2 

920123 

920176 

920228 

920280 

920332 

920384 

920436 

920489 

0541 

0593 

62 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

52 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

62 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

62 

6 

2206 

2258 

2310 

2362 

2414 

2468 

2518 

2570 

2622 

2674 

62 

7 

2725 

2777 

2829 

2881 

2933 

2986 

3037 

3089 

3140 

3192 

62 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

62 

9 

3762 

3814 

3S65 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

52 

840 

924279 

924331 

924383 

924434 

924486 

924538 

924589 

924641 

924693 

924744 

62 

1 

4796 

4848 

4399 

4951 

6003 

6054 

6106 

6157 

6209 

5261 

62 

2 

6312 

5364 

5415 

5467 

6518 

6570 

5621 

6673 

6725 

5776 

52 

3 

5828 

6879 

6931 

6982 

6034 

6085 

6137 

6188 

6240 

6291 

61 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

61 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

61 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

61 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

61 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

61 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

61 

850 

929419 

929470 

929521 

929572 

929623 

929674 

929725 

929776 

929827 

929879 

61 

1 

9930 

9981 

930032 

930083 

930134 

930185 

930236 

930287 

930338 

930389 

61 

2 

930440 

930491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

61 

3 

0949 

1000 

1061 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

61 

4 

1458 

1509 

1660 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

61 

6 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

61 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

61 

7 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

61 

8 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

61 

9 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

61 

860 

934498 

934649 

934599 

934650 

934700 

934751 

934801 

934852 

934902 

934953 

60 

1 

5003 

5054 

6104 

6154 

6205 

6255 

5306 

6356 

5406 

5457 

60 

2 

5507 

5558 

6608 

6658 

5709 

6759 

6809 

6860 

5910 

6960 

60 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

60 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

60 

6 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

60 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

g 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

60 

9 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

60 

870 

939519 

939569 

939619 

939669 

939719 

939769 

939819 

939869 

939918 

939968 

50 

1 

940018 

940068 

9401  18 

940163 

940218 

940267 

940317 

940367940417 

940467 

60 

'* 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865  0915 

0964 

60 

2 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362   1412 

1462 

60 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859  1909 

1958 

60 

5 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355;  2405 

2455 

60 

6 

2504 

2554  2603 

2653 

2702 

2752 

2801 

2851  i  2901 

2950 

60 

j 

3000 

30*9  3099 

3148!  319S 

3217 

3297 

3346  3396 

3445 

49 

8 

3495 

3544 

3598;  3643 

3692 

3742 

3791 

3841  i  3890 

3939 

49 

9 

3989 

4038 

4083 

4137 

4186 

4236 

.  4285 

4335  4384 

4433 

49 

No 

0 

1 

3 

~3~!^~ 

5 

6 

7    8 

9 

IMS 

TABLE    XIV.       LOGARITHMS    OF   NUMBERS. 


235 


No.'  0 

1 

3 

3 

4 

5    6 

7 

8 

9 

Diff. 

880 

941483 

J44532 

944531 

944631 

944680 

944729  944779  944323 

944877 

944927 

49 

1 

4976 

5025 

5074 

5124 

5173 

5222  5272 

5321 

5370 

5419 

49 

2 

5469 

6518 

5567 

5616 

5665 

5715;  5764 

5813 

5862 

5912 

49 

3 

5981 

6010 

6059 

6108 

6157 

6207!  6256 

6305 

6354 

6403 

49 

4 

6452 

6501 

6551 

6600 

6649 

6698  !  6747 

6796 

6845 

6894 

49 

5 

6943 

6992 

7041 

7090 

7140 

71891  7238 

7287 

7336 

7385 

49 

6 

7434 

7483 

7532 

7581 

7630 

7679  7728 

7777 

7826 

7876 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

8 

8413 

.8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

49 

9  8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

S90 

949390 

949439 

949488 

949536 

949585 

949634 

949683 

949731 

949780 

949829 

49 

I 

9878 

9926 

9975 

950024 

950073 

950121 

950170 

950219 

950267 

950316 

49 

2 

950365 

950414 

950462 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

49 

3 

0851 

0900 

0949 

0997 

1046 

1095 

114i 

1192 

1240 

1289 

49 

4 

1338 

13S6 

1435 

1483 

1532 

1580|  1629 

1677 

1726 

1775 

49 

5 

1823 

1872 

1920 

1969 

2017 

2066  '2114 

2163 

2211 

2260 

49 

6 

2308 

2356 

2405 

2453 

2502 

2550  2599 

2647 

2696 

2744 

43 

7 

2792 

2841 

2889 

2938 

2986 

3034  3083 

3131 

3180 

3223 

48 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566, 

3615 

3663 

3711 

48 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4116 

4194 

48 

900 

954243 

954291 

954339 

954387 

954435 

954484 

954532 

954580 

951628 

954677 

48 

i 

4725 

4773 

4821 

4869 

4918 

4966  5014 

5062 

5110 

5158 

48 

2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

6640 

I* 

3|  5688 

5736 

5784 

5832 

5380 

5928 

5976 

6024 

6072 

6120 

48 

4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

48 

5 

0049 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

6 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

48 

7  7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

48 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

9 

8664 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

48 

910 

959041 

959089 

959137 

959185 

959232 

959280 

959328 

959375 

959423 

959471 

48 

1 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

48 

2 

9995 

960042 

960090 

960138 

960185 

960233  960230 

960328 

960376 

960423 

48 

3  !  960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

48 

4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

48 

5 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

47 

6 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

47 

7 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

47 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47 

920 

963788 

963835 

963882 

963929 

963977 

964024 

964071 

964118 

964165 

964212 

47 

1 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

2 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

6108 

6155 

47 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

6678 

5625 

47 

4 

5672 

5719 

5766 

5313 

5860 

5907 

5954 

6001 

6048 

6095 

47 

5 

6142 

6189 

6236 

6233 

6329 

6376 

6423 

6470 

6517 

6564 

47 

6 

6611 

6658 

6705 

6752 

6799 

6345 

6892 

6939 

6986 

7033 

47 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

47 

8 

7548 

7595 

7642 

7638 

7735 

7782 

7829 

7875 

7922 

7969 

47 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

47 

930 

968483 

968530 

968576 

963623 

968670 

963716 

968763 

968310 

968856 

968903 

47 

1 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

47 

2 

9416 

9463 

9509 

9556 

9602 

9649 

9695  9742 

9789 

9835 

47 

3 

9882 

9928 

9975 

970021 

970063 

970114 

970161  970207 

970254 

970300 

47 

4 

970347 

970393 

970440 

04-36 

05331  0579 

0626  0672 

0719 

0765 

46 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090  1137 

1183 

1229 

46 

6 

1276 

1322 

1369 

1415 

1461 

1508 

1554   1601 

1647 

1693 

46 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018  2064 

2110 

2157 

46 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

9 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

46 

No. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Diff. 

236 


TABLE   XIV.       LOGARITHMS    OF   NUMBERS. 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

940 

973128 

973174 

973220 

973266 

973313 

973359 

973405 

973451 

973497 

97'3543 

46 

1 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

46 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

43?'4 

4420 

44G6 

46 

3 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

46 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

5 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

6 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

8 

6808 

6854 

6900 

6946 

6992 

703?' 

7083 

7129 

7175 

7220 

46 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

950 

977724 

977769 

977815 

977861 

977906 

977952 

977998 

978043 

978089 

978135 

46 

1 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

46 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

3 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

46 

4 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

5 

980003 

980049 

980094 

980140 

980185 

980231 

980276 

980322 

980367 

980412 

45 

6 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

45 

7 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

8 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

900 

982271 

982316 

982362 

982407 

982452 

982497 

982543 

982588 

982633 

982678 

45 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

45 

2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

45 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

45 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

5 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

45 

8 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

986772 

986817 

986861 

986906 

986951 

986996 

987040 

987085 

987130 

987175 

45 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

45 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8435 

8470 

8514 

45 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9364 

9405 

45 

6 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

44 

7 

9895 

9939 

9983 

990028 

990072 

990117 

990161 

990206 

990250 

990294 

44 

8 

990339 

990383 

990428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

44 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

44 

980 

991226 

991270 

991315 

991359 

991403 

991448 

991492 

991536 

991580 

991625 

44 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

44 

<•> 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

44 

4 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

6 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

7 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

44 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

44 

990 

995635 

995679 

995723 

995767 

995811 

995854 

995898 

995942 

995986 

996030 

44 

1 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

2 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

3 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

44 

4 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

5 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

44 

6 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

44 

7 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

44 

8 

9131 

9174 

9218 

9261 

9305 

9348 

.  9392 

9435 

9479 

9522 

44 

9 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

No. 

O 

1 

<j 

3 

4 

5 

6 

7 

8 

9 

Diff. 

TABLE     XV. 

LOGARITHMIC   SINES,    COSINES,    TANGENTS, 
AND   COTANGENTS. 


238  TABLE    XV.       LOGARITHMIC    SINES, 


NOTE. 

THE  table  here  given  extends  to  minutes  only.  The  usual 
method  of  extending  such  a  table  to  seconds,  by  proportional 
parts  of  the  difference  between  two  consecutive  logarithms,  is  ac- 
curate enough  for  most  purposes,  especially  if  the  angle  is  not 
very  small.  When  the  angle  is  very  small,  and  great  accuracy  is 
required,  the  following  method  may  be  used  for  sines,  tangents, 
and  cotangents. 

I.  Suppose  it  were  required  to  find  the  logarithmic  sine  of  5'  24", 
By  the  ordinary  method,  we  should  have 

log.  sin.  5'         =  7.1626961 
diff.  for  24"      =       31673 


log.  sin.  5'  24"  =  7.194369 

The  more  accurate  method  is  founded  on  the  proposition  in  Trigo- 
nometry, that  the  sines  or  tangents  of  very  small  angles  are  pro- 
portional to  the  angles  themselves.  In  the  present  case,  there- 
fore, we  have  sin.  5' :  sin.  5'  24"  =  5' :  5'  24"  =  300"  :  324".  Hence 

324  sin  5' 

sin.  5'  24"  =  —       '     ,  or  log.  sin.  5'  24"  =  log.  sin.  5'  +  log.  324  — 
oUU 

log.  300.  The  difference  for  24''  will,  therefore,  be  the  difference 
between  the  logarithm  of  324  and  the  logarithm  of  300.  The 
operation  will  stand  thus  :— 

log.  324  =  2.510545 

log.  300  =  2.477121 

diff.  for  24"      =      33424 
log.  sin.  5'         =  7.162696 

log.  sin.  5'  24"  =  7.196120 

Comparing  this  value  with  that  given  in  tables  that  extend  to 
seconds,  we  find  it  exact  even  to  the  last  figure. 

TI.  Given  log.  sin.  A  —  7.004438  to  find  A.     The  sine  next  less 
than  this  in  the  table  is  sin.  3'  =  6.940847.     Now  we  have  sin.  3' : 

sin.  ^1  =  3:^.    Therefore,  A  =  ~t  "i^~  ,  or  log.  A  =  log.  3  +  • 


COSINES,    TANGENTS,    AND    COTANGENTS.  239 

log.  sin.  A  —  log.  sin.  3f.  Hence  it  appears,  that,  to  find  the  loga- 
rithm of  A  in  minutes,  we  must  add  to  the  logarithm  of  3  the 
difference  between  log.  sin.  A  and  log.  sin.  3'. 

log.  sin.  A  =  7.004438 
log.  sin.  3'  =  6.940847 

63591 
log.  3          =  0.477121 

A  =  3.473       0.540712 

or  A  =  3'  28.38".  By  the  common  method  we  should  have  found 
4  =  3'  30.54". 

The  same  method  applies  to  tangents  and  cotangents,  except 
that  in  the  case  of  cotangents  the  differences  are  to  be  subtracted. 


*  #  *  The  radius  of  this  table  is  unity,  and  the  characteristics  9r 
8,  7,  and  6  stand  respectively  for  —  1,  —2,  —3,  and  —4. 


240           TABLE  XV.   LOGAKITHMIC  SINES, 
DC                                                1T90 

M. 

Sine. 

D.I  . 

Cosine 

D.  1". 

Tang. 

D.  1". 

Cotang. 

If. 

0 

Inf.  neg. 

0.000000 

Inf.  neg. 

Infinite. 

60 

1 
2 
3 
4 
5 
6 
7 
8 
9 

6.463726 
.764756 
.940847 
7.065786 
.162696 
.241877 
.308824 
.366816 
.417968 

5017.17 

2934.85 
2082.31 
615.17 
319.69 
1  15.78 
966.53 
852.54 
762.62 

.000000 
.000000 
.000000 
.000000 
.000000 
9.999999 
999999 
.999999 
.999999 

.00 
.00 
.00 
.00 
.00 
.00 
.00 
.01 
.01 

6.463726 
.764756 
.940847 
7.065786 
.162696 
.241878 
.308825 
.366817 
.417970 

6017.17 
2934.85 
2082.31 
615.17 
319.69 
115.78 
966.54 
852.55 
762.63 

3.536274 
.235244 
.059153 
2.934214 
.837304 
.758122 
.691175 
.633183 
.682030 

59 
58 
67 
56 
55 
54 
63 
52 
61 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

7.463726 
.505118 
.642906 
.577668 
.609853 
.639816 
.667845 
.694173 
.718997 
.742478 

689.88 
629.81 
579.37 
536.41 
499.38 
467.14 
438.81 
413.72 
391.35 
371.27 

9.999998 
.999998 
.999997 
.999997 
.999996 
.999996 
.999995 
.999995 
.999994 
.999993 

.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 

7.463727 
.505120 
.542909 
.677672 
.609857 
.639820 
.667849 
.694179 
.719003 
.742484 

689.88 
629.81 
579.37 
536.42 
499.39 
467.15 
438.82 
413.73 
391.36 
371.28 

2.536273 
.494880 
.457091 
.422328 
.390143 
.360180 
.332151 
.305821 
.280997 
.257516 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 

24 
25 
26 
27 
28 
29 

7.764754 

.785943 
.806146 
.825451 
.843934 
.861662 
.878695 
.895085 
.910879 
.926119 

353.15 
336.72 
321.75 
308.05 
295.47 
283.88 
273.17 
263.23 
253.99 
245.38 

9.999993 
.999992 
.999991 
.999990 
.999989 
.999989 
.999988 
.999987 
.999986 
.999985 

.01 
.01 
.01 
.01 
02 
.02 
.02 
02 
.02 
.02 

7.764761 
.785951 
.806155 
.825460 
.843944 
.861674 
.878708 
.895099 
.910894 
.926134 

353.16 
336.73 
321.76 
308.07 
295.49 
283.90 
273.18 
263.25 
254.01 
245.40 

2.235239 
.214049 
.193845 
.174540 
.156056 
.138326 
.121292 
.104901 
.089106 
.073866 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

7.940842 
.955082 
.968870 
.982233 
.995198 
8.007787 
.020021 
.031919 
.043501 
.054781 

237.33 
229.80 
222.73 
216.08 
209.81 
203.90 
198.31 
193.02 
188.01 
18325 

9.999983 
.999982 
.999981 
.999980 
.999979 
.999977 
.999976 
.999975 
.999973 
.999972 

.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 

7.940858 
.955100 
.968889 
.982253 
.995219 
8.007809 
.020044 
.031945 
.043527 
.054809 

237.35 
229.82 
222.76 
216.10 
209.83 
203.92 
198.33 
193.05 
188.03 
183.27 

2.059142 
.044900 
.031111 
.017747 
.004781 
1.992191 
.979956 
.968055 
.956473 
.945191 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

8.065776 
.076500 
.086965 
.097183 
.107167 
.116926 
.126471 
.135810 
.144953 
.153907 

178.72 
174.42 
170.31 
166.39 
162.65 
159.08 
155.66 
152.38 
149.24 
146  22 

9.999971 
.999969 
.999968 
.999966 
.999964 
.999963 
.999961 
.999959 
.999958 
.999956 

.02 
.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 

8.065806 
.076531 
.086997 
.097217 
.107203 
.116963 
.126510 
.135851 
.144996 
.153952 

178.75 
174.44 
170.34 
166.42 
162.68 
159.11 
155.69 
152.41 
149.27 
146.25 

1.934194 
.923469 
.913003 
.902783 
.892797 
.883037 
.873490 
.864149 
.855004 
.846048 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.162681 
.171280 
.179713 
.187985 
.196102 
.204070 
.21  1895 
.219581 
.227134 
.234557 
.241855 

143.33 
140.54 
137.86 
135.29 
132.80 
130.41 
128.10 
125.87 
123.72 
121.64 

9.999954 
.999952 
.999950 
.999948 
.999946 
.999944 
.999942 
.999940 
.999938 
.999936 
.999934 

.03 
.03 
.03 
.03 
.03 
.03 
.03 
.04 
.04 
.04 

8.162727 
.171328 
,179763 
.188036 
.196156 
.204126 
.211953 
.219641 
.227195 
.234621 
.241921 

143.36 
140.57 
137.90 
135.32 
132.84 
130.44 
128.14 
125.91 
123.76 
121.68 

1.837273 
.828672 
.820237 
811964 
.803844 
.795874 
.788047 
.780359 
.772805 
.765379 
.758079 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.  I". 

Sine. 

D.  1". 

Coteiig. 

D.  1". 

Tang. 

M. 

COSINES,    TANGENTS,    AND  COTANGENTS. 


241 


M. 

Sine 

D.  1'  . 

Cceine. 

D  1". 

Tang. 

D.  1". 

Cotaiig. 

M. 

0 
1 
2 
3 
i 
6 
6 
7 
8 
9 

8.241855 
.249033 
.256094 
.263042 
.269881 
.276614 
.283243 
.289773 
.296207 
.302546 

119.63 
117.69 
115.80 
113.98 
11221 
110.50 
108.83 
107.22 
105.66 
104.13 

9.999934 
.999932 
.999929 
.999927 
.999925 
.999922 
.999920 
.999918 
.999915 
.999913 

.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 

8.241921 
.249102 
.256165 
.263115 
.269956 
.276691 
.283323 
.289856 
.29S292 
.302634 

119.67 
117.72 
115.84 
114.02 
112.25 
110.54 
108.87 
107.26 
105.70 
104.18 

1.758079 
.750898 
.743335 
.736885 
.730044 
.723309 
.716677 
710144 
.703708 
.697366 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

8.308794 
.314954 
.321027 
.327016 
.332924 
.338753 
344504 
.350181 
.355783 
.361315 

102.66 
101.22 
99.82 
98.47 
97.14 
95.86 
94.60 
93.38 
92.19 
91.03 

!  999907 
.999905 
.999902 
.999899 
.999897 
.999894 
.999891 
.999888 
.999885 

.04 
.04 
.04 
.05 
.05 
.05 
.05 
.05 
.05 
.05 

8.308884 
.315046 
.321122 
.327114 
.333025 
.338856 
.344610 
.350289 
.355895 
.361430 

102.70 
101.26 
99.87 
98.51 
97.19 
95.90 
94.65 
93.43 
92.24 
91.08 

1.691116 
.684954 
.678878 
672886 
666975 
.661144 
.655390 
.649711 
.644105 
.638570 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
81 

8.366777 
.372171 

89.90 

9.999882 
.999879 

.05 

8.366895 
.372292 

89.95 

1.633105 

.627708 

40 
39 

22 
23 
24 
26 
86 
27 
28 
29 

.377499 
.382762 
.387962 
.393101 
.398179 
.403199 
408161 
.413068 

87.72 
86.67 
85.64 
84.64 
83.66 
82.71 
81.77 
80.86 

.999876 
.999873 
.999870 
.999867 
.999864 
.999861 
.999858 
.999854 

.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 

.377622 
.382889 
.388092 
.393234 
.398315 
.403338 
.408304 
.413213 

87.77 
86.72 
85.70 
84.69 
83.71 
82.76 
81.82 
80.91 

.622378 
.617111 
.611908 
.606766 
.601685 
.596662 
.591696 
.586787 

38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
35 
37 
38 
39 

8.417919 
.422717 
.427462 
.432156 
.436800 
.441394 
.445941 
.450440 
.454893 
.459301 

79.96 
79.09 
78.23 
77.40 
76.58 
75.77 
74.99 
74.22 
73.47 
72.73 

9.999851 
.999848 
.999844 
.999841 
.999838 
.999834 
.999831 
.999827 
.999824 
.999820 

.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 

8.418068 
.422869 
.427618 
.432315 
.436962 
.441560 
.446110 
.450613 
.455070 
.459481 

80.02 
79.14 
78.29 
77.45 
76.63 
75.83 
75.05 
74.28 
73.53 
72.79 

1.581932 
.677131 
.572382 
.567685 
.563038 
658440 
.653890 
.549387 
.544930 
.540519 

30 
29 
23 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 

46 

47 

43 
19 

8.463665 
.467985 
.472263 
.476498 
.480693 
.484848 
488963 
.493040 
,497078 
.501080 

72.00 
71.29 
70.60 
69.91 
69.24 
68.59 
67.94 
67.31 
66.69 
66.08 

9.999S16 
.999813 
.999809 
.999805 
.999801 
.999797 
.999794 
.999790 
.999786 
.999782 

.06 
.06 
.06 
.06 
.06 
.06 
.07 
.07 
.07 
.07 

8.463849 
.468172 
.472454 
.476693 
.480892 
.485050 
.489/70 
.493250 
.497293 
.501298 

72.06 
71.35 
70.66 
69.98 
69.31 
68.65 
68.01 
67.38 
66.76 
66.15 

1.536151 

.531828 
.527546 
.523307 
.519108 
.514950 
.510830 
.506750 
.502707 
.498702 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
6? 
54 
55 
56 
57 
58 
59 
60 

8505045 
.508974 
.512867 
.516726 
.520551 
.524343 
.528102 
.531828 
.535523 
.539186 
.542819 

65.48 
64.89 
64.32 
63.75 
63.19 
62.65 
62.11 
61.58 
61.06 
60.55 

9.999778 
.999774 
.999769 
.999765 
.999761 
.999757 
.999753 
.999743 
.999744 
.999740 
.999735 

.07 
.07 
.07 

.07 
.07 
.07 
.07 
.07 
.07 
.07 

8.605267 
.509200 
.513098 
.516961 
.520790 
.524586 
.528349 
.532080 
.535779 
.539447 
.543084 

65.55 
64.96 
64.39 
63.82 
63.26 
62.72 
62.18 
61.65 
61.13 
60.62 

1.494733 
.490800 
.486902 
.483039 
.479210 
.475414 
.471651 
.467920 
.464221 
.460553 
.456916 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0 

M. 

Cortne. 

D.r. 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

17 


88* 


242           TABLE  XV.   LOGARITHMIC  SINES, 
8°                                              177 

M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1«. 

Cotang. 

M. 

0 
1 
2 
3 
4 
6 
6 
7 
8 
9 

8.542819 
.546422 
.549995 
.553539 
.557054 
.560540 
.663999 
.567431 
670836 
.574214 

60.04 
69.55 
59.06 
68.58 
58.11 
57.65 
57.19 
56.74 
56.30 
55.87 

9.999735 
999731 
.999726 
.999722 
.999717 
.999713 
.999708 
.999704 
999699 
.999694 

.07 
.07 
.08 
.08 
.08 
.08 
.08 
.08 
.08 
.08 

8.543084 
.546691 
.550268 
.553817 
.557336 
.560828 
.564291 
.667727 
.671137 
.574520 

60.12 
59.62 
59.14 
58.66 
58.19 
57.73 
57.27 
56.82 
56.38 
55.95 

1.456916 
.453309 
.449732 
.446183 
.442664 
.439172 
.435709 
,432273 
.428863 
.425480 

60 
59 
58 
57 
56 
55 
64 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

8.577566 
.580892 
.584193 
.687469 
.590721 
.593948 
.597152 
.600332 
.603489 
.606623 

55.44 
55.02 
54.60 
54.19 
53.79 
53.39 
53.00 
52.61 
52.23 
51.86 

9.999689 
.999685 
.999680 
.999675 
.999670 
.999665 
.999660 
.999655 
.999650 
999646 

.08 
.08 
.08 
.08 
.08 
.08 
.08 
.08 
.08 
.09 

8.577877 
.581208 
.684514 
.587795 
.591051 
.594283 
.597492 
.600677 
.603839 
.606978 

55.52 
55.10 
54.68 
54.27 
53.87 
53.47 
53.08 
52.70 
52.32 
51.94 

1.422123 

.418792 
.415486 
.412205 
.408949 
.405717 
.402508 
.399323 
.396161 
.393022 

50 
49 
48 
47 
46 
45 
44 
43 
12 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

8.609734 
.612823 
.615891 
.618937 
.621962 
.624965 
.627948 
.63091  1 
.633854 
.636776 

51.49 
51.12 
50.77 
50.41 
50.06 
49.72 
49.38 
49.04 
48.71 
48.39 

9.999640 
999635 
999629 
999624 
999619 
999614 
.999608 
999003 
999597 
999592 

.09 
.09 
.09 
.09 
.09 
.09 
.09 
.09 
.09 
.09 

8.610094 
.613189 
.616262 
.619313 
.622343 
.625352 
.628340 
.631308 
.634256 
.637184 

51.58 
51.21 
50.85 
50.50 
50.15 
49.81 
49.47 
49.13 
48.80 
48.48 

1.389906 
.386811 
.383738 
380687 
377657 
374648 
.371660 
.36.8692 
.365744 
.362816 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 

32 
33 
34 
35 
36 
37 
38 
39 

8.639680 
.642563 
645428 
.648274 
.651102 
.653911 
.656702 
.659475 
.662230 
.664968 

48.06 
47.75 
47.43 
47.12 
46.82 
46.52 
46.22 
45.93 
45.63 
45.35 

9.999586 
999581 
999575 
999570 
999564 
999558 
999553 
999547 
999541 
999535 

.09 
.09 
.09 
.09 
.09 
.10 
.10 
.10 
.10 
.10 

8.640093 
.642982 
.645853 
.648704 
.651537 
.654352 
.657149 
.659928 
.662689 
.665433 

48.16 
47.84 
47.53 
47.22 
46.91 
46.61 
46.31 
46.02 
45.73 
45.45 

1.359907 
.357018 
.354147 
.351296 
.348463 
.345648 
.342851 
.340072 
.33731  1 
.334567 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 

42 
43 
44 
45 
46 
47 
48 
49 

8.667G89 
.670393 
.673080 
.675751 
.678405 
.681043 
.683665 
.686272 
.688863 
.691438 

45.07 
44.79 
44.51 
44.24 
43.97 
43.70 
43.44 
43.18 
42.92 
42.67 

9.999529 
999524 
.999518 
.999512 
.999506 
.999500 
.999493 
.999487 
.999481 
.999475 

.10 

.10 
.10 
.10 
.10 
.10 
10 
.10 
.10 
.10 

8.668160 
.670870 
.673563 
.676239 
.678900 
.681544 
.684172 
.686784 
.689381 
.691963 

45.16 

44.88 
44.61 
44.34 
44.07 
43.80 
43.54 
43.28 
43.03 
42.77 

1.331840 
.329130 
.326437 
•323761 
.321100 
.318456 
.315828 
.313216 
.310619 
.308037 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.693998 
.696543 
.699073 
.701589 
.704090 
.706577 
.709049 
.711507 
.713952 
716383 
.718800 

42.42 
42.17 
41.93 
41.68 
41.44 
41.21 
40.97 
40.74 
40.51 
40.29 

9.999469 
.999463 
.999456 
.999450 
.999443 
.999437 
.999431 
.999424 
.999418 
.999411 
.9994-^4 

.10 

.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 

8.694529 
.697081 
.699617 
.702139 
.704646 
.707140 
.709618 
.712083 
.7H534 
.710972 
.719396 

42.52 
42.28 
42.03 
41.79 
41.65 
41.32 
41.08 
40.85 
40.62 
40.40 

1.305471 
.302919 
.300383 
.297881 
.295354 
.292860 
.290382 
.287917 
.285466 
.283028 
.280604 

10 
9 
8 
7 
6 
6 
4 
3 
2 
I 
0 

M. 

.'_  •-  ' 

Cosine. 

—  ^= 

D.  1". 

Sine    n.  l». 

Cotaiig. 

D.  1". 

Tatg. 

M. 

COSINES,    TANGENTS,    AND   COTANGENTS. 


243 
1760 


M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 

2 
3 
4 

5 
8 

7 
8 
9 

8.718800 
.721204 
.723595 
.725972 
.723337 
.730688 
.733027 
.735354 
.737667 
.739969 

40.06 
39.34 
39.62 
39.41 
39.19 
38.93 
33.77 
38.57 
38.36 
38.16 

9.999404 
.999398 
.999391 
.999384 
.999378 
.999371 
.999364 
.999357 
.999350 
999343 

.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.12 
.12 

8.719396 
.721806 
.724204 
.726588 
.728959 
.731317 
.733663 
.735996 
.738317 
.740626 

40.17 
39.95 
39.74 
39.52 
39.31 
39.10 
38.89 
38.68 
38.48 
38.27 

1.280604 
.278194 
.275796 
.273412 
.271041 
.268683 
.266337 
.264004 
.261683 
.259374 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 

3.742259 
.744536 
.746802 
.749055 
,751297 
.753523 
.755747 
.757955 
.760151 

37.96 
37.76 
3756 
37.37 
37.17 
36.98 
36.80 
36.61 

9.999336 
.999329 
.999322 
.999315 
.999308 
.999301 
.999294 
.999237 
.999279 

.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 

8.742922 
.745207 
.747479 
.749740 
.751989 
.754227 
.756453 
.758668 
.760872 

38.07 
37.88 
37.68 
37.49 
37.29 
37.10 
36.92 
36.73 

OC  EC 

1.257078 
.254793 
.252521 
.250260 
.248011 
.245773 
.243547 
.241332 
.239128 

50 
49 
48 
47 
46 
45 
44 
43 
42 

19 

.762337 

36.24 

999272 

.12 

.763065 

36.36 

.236935 

41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

8.764511 
.766675 

.768828 
.770970 
.773101 
.775223 
.777333 
.779434 
.781524 
.783605 

36.06 
35.88 
35.70 
35.53 
35.35 
35.18 
35.01 
34.84 
34.67 
34.51 

»  .999265 
.999257 
.999250 
.999242 
.999235 
.999227 
999220 
999212 
999205 
999197 

.12 
.12 
.12 
.12 
.13 
.13 
.13 
.13 
.13 
.13 

8.765246 
.767417 
.769578 
.771727 
.773866 
.775995 
778114 
.780222 
.782320 
.784408 

36.18 
36.00 
35.83 
35.65 
35.48 
35.31 
35.14 
34.97 
34.80 
34.64 

1.234754 
.232583 
.230422 
.228273 
.226134 
.224005 
.221886 
.219778 
.217630 
.215592 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

8.785675 
.787736 
.789787 
.791828 
.793859 
.795881 
.797894 
.799897 
.801892 
.803876 

34.34 
34.18 
34.02 
33.86 
33.70 
33.54 
33.39 
33.23 
33.08 
32.93 

9.999189 
.999181 
999174 
999166 
.999158 
.999150 
.999142 
.999134 
.999126 
.999118 

.13 
13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 

8.786486 
.788554 
790613 
792662 
.794701 
.796731 
.793752 
.800763 
.802765 
.804758 

34.47 
34.31 
34.15 
33.99 
33.83 
33.68 
33.52 
33.37 
33.22 
33.07 

1.213514 
.211446 
.209387 
.207338 
205299 
.203269 
.201248 
.199237 
.197235 
.195242 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

8.805852 
.807819 
.809777 
.811726 
.813667 
.815599 
.817522 
.819436 
.821343 
.823240 

32.78 
32.63 
32.49 
32.34 
32.20 
32.05 
31.91 
31.77 
31.63 
31.49 

9.999110 
.999102 
.999094 
.999036 
.999077 
.999069 
.999061 
.999053 
.999044 
.999036 

.14 
.14 
14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 

8.806742 
.808717 
.810683 
.812641 
.814589 
.816529 
.818461 
.820384 
.822298 
.824205 

32.92 
32.77 
32.62 
32.48 
32.33 
32.19 
32.05 
31.91 
31.77 
31.63 

1.193258 
.191283 
.189317 
.187359 
.185411 
.183471 
.181539 
.179616 
.177702 
.175795 

20 

19 
18 
17 
16 

14 
13 
12 
11  i 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.82513T 
.827011 

.828884 
.830749 
.832607 
.834456 
.836297 
.833130 
.839956  ! 
.841774  ! 
.843535  ; 

31.36 
31.22 
31.03 
30.95 
30.82 
30.69 
3056 
30.43 
30.30 
30.17 

9.999027 
.999019 
.999010 
.999002 
.993993 
.993934 
.993976 
.998967 
.993953 
.998950 
.998941 

.14 
.14 
.14 
.14 
.14 
.14 
.15 
15 
.15 
.15 

8.826103 
.827992 
.829874 
831748 
.833613 
.835471 
.837321 
.839163 
.840993 
.842825 
.844644 

31  50 
S1.36 
31.23 
31.09 
30.96 
30.83 
30.70 
30.57 
30.45 
30.32 

1.173897 
.172008 
.170126 
.168252 
.166387 
.164529 
.162679 
.160837 
159002 
.157175 
.155356 

10 

6 
5 

4 

3 
2 
1 

0 

M. 

Cosine.   D.  1" 

Sine 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

80« 


244           TABLE  XT.   LOGAKITHMIC  SINES, 
4°                                               IT&c 

M 

Bine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D  1". 

Cotang. 

M. 

0 

2 
3 

5 
6 

7 
8 
9 

8.843585 
.845387 
.847183 
.848971 
.850751 
.852525 
.854291 
.856049 
.857801 
.859546 

30.05 
29.92 
29.80 
29.68 
29.55 
29.43 
29.31 
29.19 
29.08 
28.96 

9.998941 
.998932 
.998923 
.998914 
.998905 
.998896 
.998887 
.998878 
.998869 
.998860 

.15 
.15 
.15 
.15 
.15 
.15 
15 
.15 
.15 
.15 

8.844644 
.846455 
.848260 
.850057 
.851846 
.853628 
.855403 
.857171 
.858932 
860686 

30.20 
30.07 
29.95 
29.83 
29.70 
29.58 
29.46 
29.35 
29.23 
29.11 

1.155356 
.153545 
.151740 
.149943 
.148154 
.  146372 
.144597 
.142829 
.141068 
.139314 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

8.861283 
.863014 
.864738 
.866455 
.868165 
.869868 
.871565 
.873255 
.874938 
.876615 

28.84 
28.73 
28.61 
28.50 
28.39 
28.28 
28.17 
28.06 
27.95 
27.84 

9.998851 
.998841 
.998832 
.998823 
.998813 
.998804 
.998795 
.998785 
.998776 
.998766 

.15 
.15 
.15 
.16 
.16 
16 
.16 
16 
.16 
.16 

8.862433 
.864173 
.865906 
.867632 
.869351 
.871064 
.872770 
.874469 
.876162 
.877849 

29.00 
28.88 
28.77 
28.66 
28.55 
28.43 
28.32 
28.22 
28.11 
28.00 

1.137567 
.135827 
.134094 
.132368 
.130649 
.128936 
127230 
.125531 
.123838 
.122151 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 

8.878285 
.879949 
.881607 

27.73 
27.63 

9.998757 

.998747 
.998738 

.16 
.16 

8.879529 
.881202 
.882869 

27.89 
27.79 

1.120471 
.118798 
.117131 

40 
39 
38 

23 
24 

25 
26 
27 

28 

.883258 
.884903 
.886542 
.888174 
.889801 
891421 

27.42 
27.31 
27.21 
27.11 
27.00 

.998728 
.998718 
.998708 
.998699 
.998689 
.998679 

16 
.16 
.16 
.16 
.16 

\R 

.884530 
.886185 
.887833 
.889476 
.891112 
.892742 

27.58 
27.47 
27.37 
27.27 
27.17 

97  O7 

.115470 
.113815 
.112167 
.110524 
.108888 
.107258 

37 
36 
35 
34 
33 
32 

29 

.893035 

26.80 

.998669 

.17 

.894366 

26.97 

.105634 

31 

30 
31 

8.894643 
.896246 

26.70 

9.998659 
.998649 

.17 

17 

8.895984 
.897596 

26.87 

9fl  77 

1.104016 
.102404 

30 
29 

32 
33 

34 
35 
36 
37 

38 
39 

.897842 
.899432 
.901017 
.902596 
.904169 
.905736 
.907297 
.908853 

26.51 
26.41 
26.31 
26.22 
26.12 
26.03 
25.93 
25.84 

.998639 
.998629 
.998619 
.998609 
.998599 
.998589 
.998578 
.998568 

.17 
.17 
.17 
.17 
.17 
.17 
.17 
.17 

.899203 
.900803 
.902398 
.903987 
.905570 
.907147 
.908719 
.910285 

26.67 
26.58 
26.48 
26.39 
26.29 
26.20 
26.10 
26.01 

.100797 
.099197 
.097602 
.096013 
.094430 
.092853 
.091281 
.089715 

28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 

44 
45 

46 
47 

48 
49 

8.910404 
.911949 
.913488 
.915022 
.916550 
.918073 
.919591 
.921103 
.922610 
.924112 

25.75 
25.66 
25.56 
25.47 
25.38 
25.29 
25.21 
25.12 
25.03 
24.94 

9.998558 
.998548 
.998537 
.998527 
.998516 
.998506 
.998495 
.998485 
.998474 
.998464 

.17 
.17 
.17 

.17 
.17 
.18 
.18 
.18 
.18 
.18 

8.911846 
.913401 
.914951 
.916495 
.918034 
.919568 
.921096 
.922619 
.924136 
.925649 

25.92 
25.83 
25.74 
25.65 
25.56 
25.47 
25.38 
25.29 
25.21 
25.12 

1.088154 
.086599 
.085049 
.083505 
.081966 
.080432 
.078904 
.077381 
.075864 
.074351 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.925609 
.927100 
.928587 
.930068 
.931544 
.933015 
.934481 
.935942 
.937398 
.938850 
.940296 

24.86 
24.77 
24.69 
24.60 
24.52 
24.43 
24.35 
24.27 
24.19 
24.11 

9.998453 
.998442 
.998431 
.998421 
.998410 
.998399 
.998388 
.998377 
.998366 
.998355 
.998344 

.18 
18 
.18 
18 
.18 
.18 
.18 
.18 
.18 
.18 

8.927156 
.928658 
.930155 
.931647 
.933134 
.934616 
.936093 
.937565 
.939032 
.940494 
,941952 

25.04 
24.95 

24.87 
24.78 
24.70 
24.62 
24.53 
24.45 
24.37 
24.29 

1.072844 
.071342 
.069845 
.068353 
.066866 
.065384 
.063907 
.062435 
.U60968 
.059506 
.058048 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.I'. 

Sine. 

D.  1". 

Cotang.   D.  1'  . 

Tang. 

M 

94° 


COSINES,    TANGENTS,    AND   COTANGENTS. 


245 


M 

Sine. 

D.I*. 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
6 
9 

8.940296 
.941738 
.943174 
.944606 
.946034 
.947456 
.948874 
.950287 
.951696 
.953100 

24.03 
23.95 
23.87 
23.79 
23.71 
23.63 
23.55 
23.48 
23.40 
23.32 

9.993344 
.998333 
.998322 
.998311 
.998300 
.998289 
.998277 
.993266 
.993255 
.998243 

.18 
.19 
.19 
.19 
.19 
.19 
.19 
.19 
.19 
.19 

8.941952 
.943404 
.944852 
.946295 
.947734 
.949168 
.950597 
.952021 
953441 
.954856 

24.21 
24.13 
24.05 
23.97 
23.90 
23.82 
23.74 
23.67 
23.59 
23.51 

1.058048 
.056596 
.055148 
.053705 
.052266 
.050832 
.049403 
.047979 
.046559 
.045144 

60 
59 
58 
57 
56 
55 
54 
53 
52 
61 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

8.954499 
.955894 
.957284 
.958670 
.960052 
.961429 
962801 
.964170 
.965534 
.966393 

23.25 
23.17 
23.10 
23.02 
22.95 
22.88 
22.81 
22.73 
22.66 
22.59 

9.998232 

.998220 
.998209 
.993197 
.998186 
.998174 
.998163 
.998151 
.998139 
.998123 

.19 
.19 
.19 
.19 
.19 
.19 
.19 
.20 
.20 
.20 

8.956267 
.957674 
.959075 
.960473 
.961866 
.963255 
.964639 
.966019 
.967394 
.968766 

23.44 
23.36 
23.29 
23.22 
23.14 
23.07 
23.00 
22.93 
22.86 
22.79 

1.043733 
.042326 
.040925 
.039527 
.038134 
.036745 
.035361 
.033981 
.032606 
.031234 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 

8.968249 
.969600 
.970947 
.972239 
.973628 

22.52 
22.45 
22.33 
22.31 

9.998116 

.998104 
.993092 
.998080 
.998068 

.20 
.20 
.20 
.20 

8.970133 
.971496 
.972855 
.974209 
.975560 

22.72 
22.65 
22.58 
22.51 

1.029867 
.028504 
.027145 
.025791 
.024440 

40 
39 
38 
37 
36 

25 
26 
27 
28 

.9749T.2 
.976293 
.977619 
.978941 
.980259 

22.17 
22.10 
22.03 
21.97 
21.90 

.993056 
.998044 
.998032 
.998020 
.998008 

.20 
.20 
.20 
.20 
.20 

.976906 
.978248 
.979586 
.980921 
.982251 

22.37 
22.30 
22.24 
22.17 
22.10 

.023094 
.021752 
.020414 
.019079 
.017749 

35 
34 
33 
33 
31 

30 
31 

3.981573 
.932383 

21.83 

9.W996 
.997984 

.20 

8.983577 
.984899 

22.04 

1.016423 
.015101 

30 
29 

32 
33 

.934189 
.935491 

21.77 
21.70 

.997972 
.997959 

.20 
.20 

.986217 
.987532 

21.97 
21.91 

.013783 
.012468 

28 
27 

34 
35 
36 
37 

.936789 
.938083 
.989374 
.990660 

21.57 
21.51 
21.44 

.997947 
.997935 
.997922 
.997910 

.21 
.21 
.21 

.988842 
.990149 
.991451 
.992750 

21.78 
21.71 
21.65 

.011158 
.009851 
.008549 
.007250 

26 
35 
24 
23 

38 
39 

.991943 
.993222 

21.31 
21.25 

.997897 
.997885 

.21 
.21 
.31 

.994045 
.995337 

21.52 
21.46 

.005955 
.004663 

22 
21 

40 
41 
42 
43 
44 
45 

8.994497 
.995768 
.997036 
.998299 
.999560 
9.000816 

21.19 
21.12 
21.06 
21.00 
20.94 

9.997872 
.997860 
.997847 
.997335 
.997822 
.997809 

21 
.21 
.21 
.21 
.21 

8.996624 
.997908 
.999183 
9.000465 
.001738 
.003007 

21.40 
21.34 
21.27 
21.21 
21.15 

1.003376 
.002092 
.000812 
0.999535 
.998262 
.996993 

20 
19 

18 
17 
16 
15 

46 
47 
48 
49 

.002069 
.003318 
.004563 
.005805 

20.82 
20.76 
20.70 
20.64 

.997797 
.997734 
.997771 
.997758 

.21 
.21 
.21 
.21 
.21 

.004272 
.005534 
.006792 
.008047 

21.03 
20.97 
20.91 
20.85 

.995728 
.994466 
.993208 
.991953 

14 
13 
12 
11 

60 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

9.007044 
.008278 
.009510 
.010737 
.011962 
.013182 
.014400 
.015613 
.016824 
.018031 
.019235 

20.58 
20.52 
20.46 
20.40 
20.35 
20.29 
20.23 
20.17 
20.12 
20.06 

9.997745 
.997732 
.997719 
.997706 
.997693 
.997630 
.997667 
.997654 
.997641 
.997623 
.997614 

.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 

9.009298 
.010546 
.011790 
.013031 
.014263 
.015502 
.016732 
.017959 
.019183 
.020403 
.021  620 

20.80 
20.74 
20.68 
20.62 
20.56 
20.51 
20.45 
20.39 
20.34 
20.28 

0.990702 
.989454 
.988210 
.986969 
.985732 
.984498 
.983268 
.982041 
.980817 
.979597 
.978380 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0  1 

M. 

Cosine. 

D.  1". 

Sloe. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M.  j 

840 


246           TABLE  XV.   LOGARITHMIC  SINES, 
BP                                              172P 

M. 

Sine. 

D.  1". 

Cosine. 

D  1". 

Tang. 

D.  1". 

Cotang. 

M 

0 
1 
2 
3 
4 
6 
6 
7 
8 
9 

9.019235 
.020435 
.021632 
.022825 
.024016 
.025203 
.026386 
.027567 
.028744 
.029918 

20.00 
19.95 
19.89 
19.84 
19.78 
19.73 
19.67 
19.62 
19.57 
19.51 

9.997614 
.997601 
.997588 
.997574 
.997561 
997547 
.997534 
.997520 
.997507 
.997493 

22 
22 
22 
22 
22 
22 
23 
23 
23 
23 

9.021620 
.022834 
.024044. 
.025251 
.026455 
.027655 
.028852 
.030046 
.031237 
.032425 

20.23 
20.17 
20.12 
20.06 
20.01 
19.95 
19.90 
19.85 
19.79 
19.74 

0.978380 
.977166 
.975956 
.974749 
.973545 
.972345 
.971148 
.969954 
.968763 
.967575 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.031089 
032257 
033421 
.034582 
.035741 
.036896 
.038048 
.039197 
.040342 
.041485 

19.46 
19.41 
19.36 
19.30 
19.25 
19.20 
19.15 
19.10 
19.05 
19.00 

9.997480 
.997466 
.997452 
.997439 
.997425 
.997411 
.997397 
.997383 
.997369 
.997355 

23 
23 
23 
23 
23 
,23 
.23 
.23 
.23 
.23 

9.033609 
.034791 
035969 
.337144 
.038316 
.039485 
.040651 
.041813 
.042973 
.044130 

19.69 
19.64 
19.58 
19.53 
19.48 
19.43 
19.38 
19.33 
19.28 
19.23 

0.966391 
.965209 
.964031 
.962856 
.961684 
.960515 
.959349 
.958187 
.957027 
.955870 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 

9.042625 
.043762 
.044895 
.046026 
.047154 
.048279 
.049400 

18.95 
18.90 

18.85 
18.80 
18.75 
18.70 

9.997341 
.997327 
.997313 
.997299 
.997285 
.997271 
.997257 

.23 
.23 
.24 
.24 
.24 
.24 

9.045284 
.046434 
.047582 
.048727 
.049869 
051008 
.052144 

19.18 
19.13 
19.08 
19.03 
18.98 
18.93 

0.954716 
.953566 
.952418 
.951273 
.950131 
.948992 
.947856 

40 
39 
38 
37 
36 
35 
34 

27 
28 
29 

.050519 
.051635 
.052749 

18.65 
18.60 
18.55 
18.50 

.997242 
.997228 
.997214 

.24 
.24 
.24 
.24 

.053277 
.054407 
.055535 

18.84 
18.79 
18.74 

.946723 
.945593 
.944465 

33 
32 
31 

80 
31 
32 
33 
34 
85 

9.053859 
.054966 
.056071 
.057172 
.058271 
.059367 

18.46 
18.41 
18.36 
18.31 
18.27 

9.997199 
.997185 
.997170 
.997156 
.997141 
.997127 

.24 
.24 
.24 
.24 
.24 

9.056659 
.057781 
.058900 
.060016 
.061130 
.062240 

18.70 
18.65 
18.60 
18.66 
18.61 

0.943341 
.942219 
.941100 
.939984 
.938870 
.937760 

30 
29 
28 
27 
26 
25 

36 
37 
38 
39 

.060460 
.061551 
.062639 
.063724 

18.22 
18.17 
18.13 

18.08 
18.04 

.997112 
.997098 
.997083 
.997068 

.24 

.24 
.24 
.25 
.25 

.063348 
.064453 
.065556 
.066655 

18.46 
18.42 
18.37 
18.33 

18.28 

.936652 
.935547 
.934444 
.933345 

24 
23 

22 
21 

40 
41 
42 
43 
44 
45 
46 
47 

9.064806 
.065885 
.066962 
.06S036 
.069107 
.070176 
.071242 
.072306 

17.99 
17.95 
17.90 
17.86 
17.81 
17.77 
17.72 

9.997053 
.997039 
.997024 
.997009 
.996994 
.996979 
.996964 
.996949 

.25 
.25 
.25 
.25 
.25 
.25 
.25 

9.067752 
.068846 
.069938 
.071027 
.072113 
.073197 
.074278 
.075356 

18.24 
18.19 
18.15 
18.10 
18.06 
18.02 
17.97 

0.932248 
.931154 

.930062 
928973 
.927887 
.926803 
.925722 
.924644 

20 
19 

18 
17 
16 
15 
14 
13 

48 
49 

.073366 
.074424 

17.68 
17.64 
17.59 

.996934 
.996919 

.25 
.25 
.25 

.076432 
.077505 

17.89 
17.84 

.923568 
.922495 

18 

11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

9.075480 
.076533 
.077583 
.078631 
.079676 
.080719 
.081759 
.082797 
.083832 
.084864 

17.55 
17.51 
17.46 
17.42 
17.38 
17.34 
17.29 
17.25 
17.21 

9.996904 
.996889 
.996874 
.996858 
.996843 
.996828 
.996812 
.996797 
.996782 
.996766 

.25 
.25 
.25 
.25 
.26 
.26 
.26 
.26 
.26 

9.078576 
.079644 
.080710 
.081773 
.082833 
.083891 
.084947 
.086000 
.087050 
.088098 

17.80 
17.76 
17.72 
17.67 
17.63 
17.59 
17.55 
17.51 
17.47 

0.921424 
.920356 
.919290 
.918227 
.917167 
.916109 
.915053 
.914000 
.912950 
.911902 

10 
9 

8 
7 
6 
6 
4 
3 
2 
! 

60 

.085894 

.996751 

.26 

.089144 

.910356 

0 

M 

Cosine. 

D.  1". 

Sine 

D.I. 

Cotang. 

D.  1". 

Tang. 

M 

COSINES,    TANGENTS,    AND   COTANGENTS. 


24? 


M. 

Bine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

9.085394 

1  T  1  q 

9.996751 

26 

9.089144 

1  7  *yj 

0.910856 

60 

2 
3 

4 
5 
6 
7 
8 
9 

.086922 
.087947 
.088970 
.089990 
.091008 
.092024 
.093037 
.094047 
.095056 

17.09 
17.05 
17.00 
16.96 
16.92 
16.88 
16.84 
16.80 
16.76 

.996735 
996720 
996704 
996633 
996673 
996657 
996641 
996625 
996610 

!26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 

.090187 
.091228 
.092266 
.093302 
.094336 
.095367 
.096395 
.097422 
.098446 

17.35 
17.31 
17.27 
17.23 
17.19 
17.15 
17.11 
17.07 
17.03 

.909813 
.908772 
.907734 
.906698 
.905664 
.904633 
.903605 
.902578 
.901554 

59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 

9.096062 
.097065 
.093066 
.099065 
.100062 
.101056 
.102048 
.103037 
.104025 

16.73 
16.69 
16.65 
16.61 
16.57 
16.53 
16.49 
16.46 

1  ft  49 

9.996594 
996578 
.996562 
996546 
996530 
996514 
.996498 
.996482 
.996465 

.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 

97 

9.099468 
.100487 
•101504 
.102519 
.103532 
.104542 
.105550 
.106556 
.107559 

16.99 
16.95 
16.91 
16.88 
16.84 
16.80 
16.76 
16.72 

1  A  RQ 

0.900532 
.899513 
.893-496 
.897481 
.896468 
.895458 
.894450 
.893444 
.892441 

50 
49 
48 
47 
46 
45 
44 
43 
42 

19 

.105010 

16.33 

.996449 

.27 

.108560 

16.65 

.891440 

41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.105992 
.106973 
.107951 
.108927 
.109901 
.110873 
.111842 
.112809 
.113774 
.114737 

16.34 
16.30 
16.27 
16.23 
16.19 
16.16 
16.12 
16.08 
16.05 
16.01 

9.996433 
.996417 
.996400 
.9963^ 
.996363 
.996351 
996335 
.996318 
.996302 
.996285 

.27 

.27 
.27 
.27 
.27 
.27 
.23 
.28 
.28 
.28 

9.109559 
.110556 
.111551 
.112543 
.113533 
.114521 
.115507 
.116491 
.117472 
.118452 

16.61 
16.58 
16.54 
16.50 
16.47 
16.43 
16.39 
16.36 
16.32 
16.29 

0.890441 
.839444- 
.888449 
.887457 
.886467 
.885479 
.884493 
.883509 
882523 
.881548 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

9.115698 
.116656 
.117613 
.118567 
.119519 
.120469 
.121417 
.122362 
.123306 
.124248 

15.98 
15.94 
15.90 
15.87 
15.83 
15.80 
15.76 
15.73 
15.69 
15.66 

9.996269 
.996252 
.996235 
.996219 
.996202 
.996185 
.996168 
.996151 
996134 
.996117 

.28 
.28 
.28 
.23 
.28 
.28 
.28 
28 
.28 
.28 

9.119429 
.120404 
.121377 
.122348 
.123317 
.124284 
.125249 
.126211 
.127172 
.128130 

16.25 
16.22 
16.18 
16.15 
16.11 
16.08 
16.04 
16.01 
15.98 
15.94 

0.880571 
.879596 
.878623 
.877652 
.876683 
.875716 
.874751 
.873789 
.872828 
.871870 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
15 
46 
47 
48 
49 

9.125187 
.126125 
.127060 
.127993 
.128925 
.129354 
.130781 
.131706 
.132630 
.133551 

15.62 
15.59 
15.56 
15.52 
15.49 
15.45 
15.42 
15.39 
15.35 
15.32 

9.996100 
.996083 
.996066 
.996049 
.996032 
.996015 
.995993 
.995930 
.995963 
.995946 

.28 
.28 
.28 
.29 
.29 
.29 
.29 
.29 
.29 
.29 

9.129087 
.130041 
.130994 
.131944 
.132893 
.133339 
.134784 
.135726 
.136667 
.137605 

15.91 
15.87 
15.84 
15.81 
15.77 
15.74 
15.71 
15.68 
15.64 
15.61 

0.870913 
.869959 
.869006 
.868056 
.867107 
.866161 
.865216 
.864274 
.863333 
.862395 

20 
19 
18 
17 
16 
16 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
68 
59 
60 

9.134470 
.135337 
.136303 
.137216 
.133128 
.139037 
.139944 
.140350 
141754 
142655 
.143555 

15.29 
15.26 
15.22 
15.19 
15.16 
15.13 
15.09 
15.06 
15.03 
15.00 

9.995928 
.995911 
.995891 
.995876 
.995859 
.995341 
.995823 
.995806 
.995788 
.995771 
.995753 

.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.30 

9.138542 
.139476 
.140409 
.141340 
.142269 
.143196 
.144121 
.145044 
.145966 
.146885 
.147803 

15.58 
15.55 
15.51 
15.48 
15.45 
15.42 
15.39 
15.36 
15.32 
15.29 

0.861458 
.860524 
.859591 
.858660 
.857731 
.856804 
.855879 
854956 
.854034 
.853115 
.852197 

10 
9 
8 
7 
6 
6 
4 
3 
2 

0 

M. 

Coslue. 

D.  1". 

Slue 

D  1' 

Cotaug. 

D.  1'  . 

Tang. 

248           TABLE  XV.   LOGARITHMIC  SINES, 

80                                              1T10 

M. 

Sine 

D.  1". 

Coelne. 

D.I' 

Tang. 

D.  1". 

Cotang 

M. 

0 

2 
3 
4 
5 
6 
7 
8 
9 

9  143555 
.144453 
.145349 
.146243 
.147136 
.148026 
.148915 
.149802 
.150686 
.151569 

14.97 
14.93 
14.90 
14.87 
14.84 
14.81 
14.78 
14.75 
14.72 
14.69 

9.995753 
.995735 
.995717 
.995699 
995681 
.995664 
.995646 
.995628 
.995610 
.995591 

.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 

9.  147803 
.148718 
.  149632 
.150544 
.151454 
152363 
.153269 
154174 
155077 
.155978 

15.26 
15.23 
15.20 
15.17 
15.14 
15.11 
15.08 
15.05 
15.02 
14  99 

0.852197 
.851282 
.850363 
.849456 
.848546 
.847637 
.846731 
.845826 
.844923 
.844022 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.152451 
.153330 
154208 
.155083 
.155957 
,156830 
.157700 
.158569 
.159435 
.160301 

14.66 
14.63 
14.60 
14.57 
14.54 
14.51 
14.48 
14.45 
14.42 
14.39 

9.995573 
.995555 
.995537 
.995519 
.995501 
.995482 
.995464 
.995446 
.995427 
.995409 

.30 
.30 
.30 
.30 
.30 
31 
.31 
.31 
.31 
.31 

9.156877 
.157775 
.158671 
.159565 
.160457 
.161347 
.162236 
.163123 
.164008 
.164892 

14.96 
14.93 
14.90 
14.87 
14.84 
14.81 
14.78 
14.75 
14.73 
14  70 

0.843123 

.842225 
.841329 
.840435 
.839543 
.838653 
.837764 
.836877 
835992 
.835108 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.161164 
.162025 
.162885 
.163743 
.164600 
.165454 
.166307 
.167159 
168008 
.168856 

14.36 
14.33 
14.30 
14.27 
14.24 
14.22 
14.19 
14.16 
14.13 
14.10 

9.995390 
.995372 
.995353 
.995334 
.995316 
.995297 
.995278 
.995260 
.995241 
.995222 

.31 
.31 
31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 

9.165774 
.166654 
167532 
168409 
.169284 
.170157 
.171029 
.171899 
.172767 
.173634 

14.67 
14.64 
14.61 
14.58 
14.56 
14.53 
14.50 
14.47 
14.44 
14  42 

0.834226 
.833346 
832468 
.831591 
.830716 
.829843 
.828971 
.828101 
.827233 
.826366 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.169702 
.170547 
.171389 
.172230 
.173070 
.173908 
.174744 
.175578 
176411 
.177242 

14.07 
14.05 
14.02 
13.99 
13.96 
13.94 
13.91 
13.88 
13.85 
13.83 

9.995203 
.995184 
.995165 
.995146 
.995127 
.995108 
.995089 
.995070 
..995051 
.995032 

.31 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
32 

9.174499 
.175362 
.176224 

.177084 
.177942 
.178799 
.179655 
.180508 
.181360 
.182211 

14.39 
14.36 
14.33 
14.31 
14.28 
14.25 
14.23 
14.20 
14.17 
14  15 

0.825501 
.824688 
.823776 
.822916 
.822058 
.821201 
.820345 
.819492 
.818640 
.81778? 

30 
29 

28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.178072 
.178900 
.179726 
.180551 
.181374 
.182196 
.183016 
.183834 
.184651 
.185466 

13.80 
13.77 
13.75 
13.72 
13.69 
13.67 
13.64 
13.61 
13.59 
13.56 

9.995013 
.994993 
.994974 
.994955 
.994935 
.994916 
.994896 
.994877 
.994857 
.S94838 

.32 
.32 
.32 
.32 
.32 
.32 
.33 
.33 
.33 
.33 

9.183059 
.183907 
.184752 
.185597 
.186439 
.187280 
.188120 
.188958 
.  189794 
.190629 

14.12 
14.09 
14.07 
14.04 
14.02 
13.99 
13.97 
13.94 
13.91 
13  89 

0.816941 
.816093 
.815248 
.814403 
.813561 
.812720 
811880 
.811  042 
.810206 
.809371 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9,186230 
.187092 
.187903 
.188712 
.189519 
.190325 
.191130 
.191933 
.192734 
.193534 
.194332 

1354 
13.51 
13.48 
13.46 
13.43 
13.41 
13.38 
13.36 
13.33 
13.31 

9.994818 
.994798 
.994779 
.994759 
.994739 
.994720 
994700 
.994680 
.994660 
.994640 
.994620 

.33 
.33 
.33 
.33 
.33 
.33 
.33 
.33 
.33 
.33 

9.191462 
.192294 
.193124 
.193953 
.194780 
.195606 
.196430 
.197253 
.198074 
.198894 
199713 

13.86 
13.84 
13.81 
13.79 
13.76 
13.74 
13.71 
13.69 
13.66 
13.64 

0.808538 
.807706 
.806876 
.806047 
.805220 
.804394 
.803570 
.802747 
.801926 
.801106 
.800287 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0 

M. 

Coelne. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Taug. 

M. 

980 


COSINES,    TANGENTS,    AND   COTANGENTS. 


ML 

Sine. 

D  1«. 

Cosine. 

D.  1". 

Timg. 

D.  1". 

Cotang. 

M. 

0 
1 

2 
3 

4 
5 
6 
7 
8 
9 

9.194332 
.195129 
.195925 
.196719 
.197511 
.198302 
.199091 
.199379 
.200666 
.201451 

13.28 
13.26 
13.23 
13.21 
13.18 
13.16 
13.13 
13.11 
13.08 
13.06 

9.994620 
.994600 
.994580 
.994560 
.994540 
.994519 
.994499 
.994479 
994459 
.994438 

.33 
.33 
.34 
34 
.34 
34 
34 
34 
.34 
34 

9.199713 
.200529 
.201345 
.202159 
.202971 
203732 
.204592 
.205400 
.206207 
.207013 

i3.62 
13.59 
13.57 
13.54 
13.52 
13.49 
13.47 
13.45 
13.42 
13.40 

0.800287 
.799471 
.798655 
.797841 
.797029 
.796218 
.795403 
.794600 
.793793 
.792937 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
!  14 
i  15 
i  16 

;  17 

!  i8 

19 

1.  202234 
.203017 
.203797 
.204577 
205&r>4 
206131 
206906 
.207679 
.203452 
.209222 

13.04 
13.01 
12.99 
12.96 
12.94 
12.92 
12.89 
12.87 
12.85 
12.82 

o  994418 
J94398 
.994377 
.994357 
994336 
994316 
994295 
994274 
.994254 
994233 

34 
34 
34 
34 

34 
34 
34 
34 
.35 
36 

9.207817 
.203619 
209420 
210220 
211018 
211815 
212611 
213405 
214193 
214989 

13.38 
13.35 
13.33 
13.31 
13.28 
13.26 
13.24 
13.21 
13.19 
13.17 

0.792183 
.791381 
.790580 
.789780 
.788982 
.788185 
.787389 
.786595 
.785802 
.78501  1 

60 
49 

48 
47 
46 
45 
44 
43 
42 
41 

;  20 

1  21 
!  22 
23 
24 
25 
26 
27 
23 
29 

9.209992 
.210760 
.211526 
.212291 
.213056 
.213818 
214579 
215338 
216097 
216S54 

12.80 
12.78 
12.75 
12.73 
12.71 
12.68 
12.66 
12.64 
12.62 
12.59 

9.994212 
994191 
.994171 
994150 
994129 
.994108 
.994037 
.994066 
994045 
.994024 

35 
35 
35 
35 
35 
.35 
35 
35 
35 
35 

9.215780 
216568 
217356 
218142 
218926 
219710 
220492 
221272 
222052 
222830 

13.15 
13.12 
13.10 
13.08 
13.06 
13.03 
13.01 
12.99 
12.97 
12.95 

0.784220 
.783432 
.782644 
.781858 
.781074 
.780290 
.779508 
.778728 
.777948 
.777170 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.217609 
.218363 
.219116 
.219368 
.220618 
.221367 
.222115 
.222.361 
.223606 
.224349 

12.57 
12.55 
12.53 
12.50 
12.48 
12.46 
12.44 
12.42 
12.39 
12.37 

9,994003 
.993932 
.993960 
.993939 
.993913 
.993397 
.993875 
.993854 
.993832 
.993811 

35 
35 
.35 
.35 
.36 
36 
.36 
.36 
.36 
.36 

9.223607 
.224382 
.225156 
.225929 
.226700 
.227471 
.223239 
.229007 
.229773 
.230539 

12.92 
12.90 
12.88 
12.86 
12.84 
12.62 
12.79 
12.77 
12.75 
12.73 

0.776393 
.775618 
.774844 
.774071 
.773300 
.772529 
.771761 
.770993 
.770227 
.769461 

30 
29 
28 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.225092 
.225833 
.226573 
.227311 
.228043 
.228784 
.229518 
.230252 
.230984 
.231715 

12.35 
12.33 
12.31 
12.29 
12.26 
12.24 
12.22 
12.20 
12.18 
12.16 

9.993789 
993768 
993746 
.993725 
.993703 
993681 
.993660 
.993633 
.993616 
993594 

.36 
.36 
.36 
.36 
.36 
36 
36 
.36 
36 
36 

9.231302 
232065 
232826 
.233586 
.234345 
235103 
.235,359 
.236614 
.237368 
.238120 

12.71 
12.69 
12.67 
12.65 
12.63 
12.60 
12.58 
12.56 
12.54 
12.52 

0.768698 
.767935 
.767174 
.766414 
.765655 
.764897 
.764141 
.763386 
.762632 
.761880 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.232444 
.233172  ! 
.233399 
.234625 
.235349 
.236073 
.236795 
.237515 
.2382:5 
.233953 
.239671 

12.14 
12.12 
12.10 
12.07 
12.05 
12.03 
12.01 
11.99 
11.97 
11  95 

9.993572 
.993550 
.993528 
.993506 
.993484 
.993462 
.993440 
.993418 
.993396 
.993374 
.993351 

37 

.37 
.37 
.37 
.37 
.37 
37 
.37 
.37 
.37 

9.238872 
.239622 
.240371 
.241118 
.241865 
.242610 
.243354 
.244097 
.244839 
.245579 
.246319 

12.50 
12.48 
12.46 
12.44 
12.42 
12.40 
12.33 
12.36 
12.34 
12.32 

0.761128 
.760378 
.759629 
.758882 
.758135 
.757390 
.756646 
.755903 
.755161 
.754421 
.753681 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0 

M. 

Cosine.  1  D.  1". 

Sloe. 

D.  1". 

Cotang. 

D.  1" 

Tang. 

M. 

800 


250 


TABLE   XV.       LOGAKITIIMIC    SINES, 


1601 


M. 

Blue. 

D.  1" 

Cosine. 

D.  1". 

Twig. 

D.  1". 

Cotang 

M 

0 

1 

8 
4 

6 
6 
7 
8 
9 

».239670 
.240386 
.241101 
.241814 
.242526 
.243237 
.243947 
.244656 
.245363 
.246069 

11.93 
11.91 
11.89 
11.87 
11.85 
11.83 
11.81 
11.79 
11.77 
11  75 

9.993351 
.993329 
.993307 
.993284 
.993262 
993240 
.993217 
.993195 
.993172 
.993149 

.37 
.37 
.37 
.37 
.37 
.37 
.38 
.38 
.38 
.38 

9.246319 
.247057 
.247794 
.243530 
.249264 
.249998 
.250730 
.251461 
.252191 
.252920 

12.30 
12.28 
12.26 
12.24 
12.22 
12.20 
12.18 
12.17 
12.15 
12.13 

0.753681 

.752943 
.752206 
.751470 
.750736 
.750002 
.749270 
.748539 
.747809 
.747080 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 

11 

9.246775 
.247478 

11.73 

9.993127 
.993104 

.38 

OQ 

9.253648 
.254374 

12.11 
12  09 

0.740352 
.745626 

50 
49 

12 
13 
14 
15 
16 
17 
18 

.248181 

.248883 
.249583 
^250282 
.250980 
.861677 
.252373 

11.71 
11.69 
11.67 
11.66 
11.63 
11.61 
11.59 

.993081 
.993059 
.993036 
.993013 
.992990 
.992967 
.992944 

.38 
38 
.38 
.38 
.38 
.38 

QQ 

.255100 
.255824 
.256547 
.257269 
.257990 
.258710 
.259429 

12.07 
12.05 
12.03 
12.01 
12.00 
11.98 
Uq/> 

.744900 
.744176 
.743453 
.742731 
.742010 
.741290 
.740571 

48 
47 
46 
45 
44 
43 
42 

19 

.253067 

11.58 
11  56 

.992921 

.38 

.260146 

11.94 

.739854 

41 

20 
21 
22 
23 
24 
26 
26 
27 
28 
29 

9.253761 
.254453 
.255144 
.265834 
.256523 
.257211 
.257898 
.258683 
.259268 
.259961 

11.54 
11.52 
11.60 
11.48 
11.46 
11.44 
11.42 
11.41 
11.39 
11  37 

9.992898 
.992875 
.992852 
.992829 
992806 
.992783 
.992759 
.992736 
,992713 
,992690 

.38 
.38 
.39 
39 
.39 
39 
39 
39 
39 
39 

9.260863 
.261578 
.262292 
.263005 
.263717 
.264428 
.265138 
.265347 
.266555 
.267261 

11.92 
11.90 
11.89 
11.87 
11.85 
11.83 
11.81 
11.79 
11.78 
11.76 

0.739137 
.738422 
.737708 
.736995 
.736283 
.735572 
.734862 
.734153 
.733445 
.732739 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
36 
36 
37 
38 
39 

9.260633 
.261314 
.261994 
.262673 
.263351 
.264027 
.264703 
.265377 
.266051 
.266723 

11.35 
11.33 
11.31 
11.30 
11.28 
11.26 
11.24 
11.22 
11.20 
11  19 

9.992666 
.992643 
.992619 
.992596 
.992572 
.992549 
.992525 
.992501 
.992478 
.992454 

39 
39 
39 
39. 
.39 
.39 
.39 
.39 
.40 
.40 

9.267967 
.268671 
.269375 
.270077 
.270779 
.271479 
.272178 
.272876 
.273573 
.274269 

11.74 
11.72 
11.70 
11.69 
11.67 
11.65 
11.64 
11.62 
11.60 
11.58 

0.732033 
.731329 
730625 
.729923 
.729221 
.728521 
.727822 
.727124 
.726427 
.725731 

30 
29 

28 
27  1 
26  I 
25  1 
24  1 
23 
22 
21 

40 
41 
42 
43 

44 
45 

46 
47 

48 
49 

9.267395 
.268065 
.268734 
.269402 
.270069 
.270735 
.271400 
.272064 
.272726 
.273388 

11.17 
11.15 
11.13 
11.12 
11.10 
11.08 
11.06 
11.05 
11.03 
11  01 

9.992430 
.992406 
.992382 
.992359 
.992335 
.992311 
.992237 
,992263 
.992239 
.992214 

.40 
.40 
.40 
40 
.40 
.40 
.40 
.40 
.40 
40 

9.274964 
.275658 
.276351 
.277043 
.277734 
.278424 
.279113 
.279801 
.230488 
.231174 

11.57 
11.55 
11.53 
11.51 
11.50 
11.48 
11.46 
11.45 
11.43 
11  41 

0.725036 
.724342 
.723649 
.722957 
.722266 
.721576 
.720887 
.720199 
.719512 
.718826 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
69 
60 

9.274049 
.274708 
.275367 
.276025 
.276631 
.277337 
.277991 
.278645 
.279297 
.279948 
.280599 

10.99 
10.98 
10.96 
10.94 
10.92 
10.91 
10.89 
10.87 
10.86 
10.84 

9.992190 
.992166 
.992142 
.992118 
.992093 
.992069 
.992044 
.992020 
.991996 
.991971 
.991947 

.40 
.40 
.40 
.41 
.41 
.41 
.41 
.41 
.41 
.41 

9.28iarj8 
.282542 
.283225 
.283907 
.284588 
.285268 
.285947 
.286624 
.287301 
.237977 
.238652 

11.40 
11.38 
11.36 
11.35 
11.33 
11.31 
11.30 
11.28 
11.26 
11.25 

0.718142 
.717458 
.716775 
.716093 
.715412 
.714732 
.714053 
.713376 
.712699 
.712023 
.711348 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0 

M. 

Cofiin«. 

D.  1". 

Sir*. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

1000 


COSINES,    TANGENTS,    AND   COTANGENTS. 


M. 

Sine 

D.1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 

9.280599 

.281248 
.281897 
.282544 

10.82 
10.81 
10.7S 

9.991947 
.991922 
.991897 
.991873 

.41 
.41 
.41 

A  1 

9.238652 
.289326 
289999 
290671 

11.23 
11.22 
11.20 

0.711348 
.710674 
710001 
709329 

60 
59 
58 
57 

4 

.283190 

.991843 

.291342 

.708658 

56 

5 

.233836 

10.76 

.991823 

292013 

707987 

55 

6 

.284480 

10.74 

.991799 

.2926S2 

.707318 

54 

7 

.285124 

10.72 

.991774 

41 

293350 

1  1.14 

706650 

53 

8 

.285766 

10.71 

.991749 

.291DI7 

705933 

52 

9 

.286408 

10.69 
10.67 

.991724 

42 

.294684 

11.11 
11.09 

.705316 

51 

1C 
11 

9.287048 

.287688 

10.66 

9.991699 
.991674 

.42 

9.295:349 
.296;)I3 

11.07 

0.704651 
.703987 

50 
49 

12 
13 
14 

(  l5 
16 
17 

18 
19 

.288326 
.288964 
.289600 
.290236 
.290370 
.291504 
.292137 
.292768 

10.64 
10.63 
10.61 
10.59 
10.58 
10.56 
10.55 
10.53 
10.51 

.991619 
.991624 
.991599 
.991574 
.991549 
.991524 
.991498 
.991473 

.42 
42 
42 
42 
.42 
.42 
.42 
.42 

.296677 
.297339 
.298001 
.293662 
299322 
299980 
300638 
.301295 

11.06 

11.04 
11.03 
11.01 
11.00 
10.98 
10.97 
10.95 
10.93 

.703323 
.702661 
.701999 
.701338 
.700678 
700020 
.699362 
.698705 

48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.293399 
.294029 
.294658 
.295236 
.295913 
.296539 
297164 
.297788 
2934  12 
.299034 

10.50 
10.48 
10.47 
10.45 
10.43 
10.42 
10.40 
10.39 
10.37 
10.36 

9.991443 
.991422 
.991397 
.991372 
.991346 
.991321 
.991295 
.991270 
.991244 
.991218 

.42 
.42 
42 
.42 
.42 
.43 
.43 
.43 
.43 
.43 

9.301951 
.3026:17 
.303261 
303914 
304567 
.305218 
305869 
.306519 
307168 
307816 

10.92 
10.90 
10.89 
10.87 
10.86 
10.84 
10.83 
10.81 
10.80 
10.78 

a  698049 
697393 
696739 
.696086 
.695433 
.694782 
.694131 
.693481 
.692832 
.692184 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

8.  299655 
.300276 
300395 
.301514 
.302132 
.302743 
303364 
303979 
.304593 
.305207 

10.34 
10.33 
10.31 
10.30 
10.28 
10.26 
10.25 
10.23 
10.22 
10.20 

9.991193 
.991167 
.991141 
.991115 
.991090 
.991064 
.991038 
.991012 
.990936 
.990960 

.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 

9.308463 
.309109 
.309754 
.310399 
.311042 
.311685 
.312327 
.312968 
.313608 
.314247 

10.77 
10.76 
10.74 
10.73 
10.71 
10.70 
10.68 
10.67 
10.65 
10.64 

0.691537 
.690891 
.690246 
.689601 
.688958 
.688315 
.687673 
.687032 
.686392 
.685753 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.305819 
.306430 
.307041 
.307650 
.303259 
.308867 
.309474 
.310080 
.310685 
.311289 

10.19 
10.17 
10.16 
10.14 
10.13 
10.12 
10.10 
10.09 
10.07 
10.06 

9.990934 
.990908 
.990882 
.990355 
.990829 
.990303 
.990777 
.990750 
.990724 
.990697 

.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 

9.314885 
.315523 
.316159 
.316795 
.317430 
.318064 
.318697 
.319330 
.319961 
.320592 

10.62 
10.61 
10.60 
10.58 
10.57 
10.55 
10.54 
10.53 
10.51 
10.50 

0.685115 

.684477 
683841 
.683205 
.682570 
.681936 
.681303 
.680670 
.680039 
.679408 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.311893 
.312495 
.313097 
.313693 
.314297 
.314897 
.315495 
.316092 
.316689 
.317284 
.317879 

10.04 
10.03 
10.01 
10.00 
9.93 
9.97 
9.96 
9.94 
9.93 
9.91 

9.990671 
.990645 
.990618 
.990591 
.990565 
.990538 
.990511 
.990485 
.990453 
.990431 
.990404 

.44 

.44 
.44 
.44 
.44 
.44 
.45 
.45 
.45 
.45 

9.321222 
.321851 
.322479 
.323106 
.323733 
.324358 
.324983 
.325607 
.326231 
326853 
.327475 

10.48 
10.47 
10.46 
10.44 
10.43 
10.41 
10.40 
10.39 
10.37 
10.36 

0.678778 
.678149 
.677521 
.676894 
676267 
.675642 
.675017 
.674393 
.673769 
673147 
.672525 

10 
9 

8 
7 
6 
5 

3 
2 

1 
0 

M. 

Octloe. 

D.  1». 

Sine, 

D.l«. 

Co  tang. 

D.  1". 

Taug. 

M. 

1010 


TABLE   XV.       LOGARITHMIC    SINES, 


16T< 


M. 

Sine 

D.  1". 

Cosine. 

D.l". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

1 

2 

9.317879 
.318473 
.319066 

9.90 
9.88 

9.990404 
.990378 
.990351 

.45 
.46 

9.327475 
.328095 
.328715 

10.35 
10.33 

0.672525 
671905 

.671285 

60 
59 

58 

3 

.319658 

9.87 

.990324 

45 

.329334 

10  31 

.670666 

57 

4 
6 

6 
7 
8 
9 

.320249 
.320840 
.321430 
.322019 
.322607 
.323194 

9.86 
9.84 
9.83 
9.81 
9.80 
9.79 
9.77 

.990297 
.990270 
.990243 
.990215 
.990188 
.990161 

'.45 
.45 
.45 
.45 
.45 
.45 

.329953 
.330570 
.331187 
.331803 
.332418 
.333033 

10^29 
10.28 
10.27 
10.25 
10.24 
10.23 

.670047 
.669430 
.668813 
.668197 
.667582 
.666967 

56 
55 
54 
53 
52 
51 

10 
11 
12 

13 
14 
16 
16 

9.323780 
.324366 
.324960 
.325534 
.326117 
.326700 
.327281 

9.76 
9.76 
9.73 
9.72 
9.70 
9.69 

n  eo 

9.990134 
.990107 
.990079 
.990052 
.990025 
.989997 
.989970 

.45 
.45 
.46 
.46 
.46 
.46 

9.333646 
.334259 
.334871 
.335482 
.336093 
.336702 
.337311 

10.21 
10.20 
10.19 
10.17 
10.16 
10.15 
10  14 

0.666354 
.665741 

.665129 
.664518 
.663907 
.663298 
.662689 

60 
49 

48 
47 
46 
45 
44 

17 

.327862 

y.oo 

Q  AA 

.989942 

IA 

.337919 

in  19 

.662081 

43 

18 
19 

.328442 
.329021 

».OO 

9.66 
9.64 

.989916 
.989887 

!46 
.46 

.338527 
.339133 

iu.  l/o 

10.11 
10.10 

.661473 
.660867 

42 
41 

20 
21 
22 

9.329599 
.330176 
.330753 

9.62 
9.61 

Q  Art 

9.989860 

.46 
.46 

9.339739 
.340344 
.340948 

10.08 
10.07 
10  06 

0.660261 
.659656 
.659052 

40 
39 
38 

23 
24 
25 
26 
27 
28 

.331329 
.331903 
.332478 
.333051 
.333624 
.334195 

y.ou 
9.58 
9.57 
9.66 
9.54 
9.53 

!  989777 
.989749 
.989721 
.989693 
.989665 
.989637 

.46 
.46 
.46 
.46 
.46 
.47 

.341552 
.342155 
.342757 
.343358 
.343958 
.344558 

lo'.os 

10.03 
10.02 
10.01 
10.00 

Q  Qft 

.658448 
.657845 
.657243 
.656642 
.656042 
.655442 

37 
36 
36 
34 
33 
32 

29 

.334767 

9.52 
9.50 

.989610 

.47 

.47 

.345157 

y.yo 
9.97 

.654843 

31 

30 
31 
32 
33 
34 
36 
36 
37 
38 
39 

9.335337 
.335906 
.336475 
.337043 
.337610 
.338176 
.338742 
.339307 
339871 
.340434 

9.49 
9.48 
9.46 
9.45 
9.44 
9.43 
9.41 
9.40 
9.39 
9.37 

9.989582 
.989553 
.989525 
.989497 
.989469 
.989441 
.989413 
.989385 
.989356 
.989328 

.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 

9.345755 
.346353 
.346949 
.347545 
.348141 
.348735 
.349329 
.349922 
.350514 
.351106 

9.96 
9.95 
9.93 
9.92 
9.91 
9.90 
9.88 
9.87 
9.86 
9.85 

0.654245 
.653647 
.653051 
.652455 
.651859 
.651265 
.650671 
.650078 
.649486 
.648894 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

4C 
41 
42 
43 
44 
45 
46 
47 

9.340996 
.341558 
.342119 
.342679 
.343239 
.343797 
.344355 
.344912 

9.36 
9.35 
9.34 
9.32 
9.31 
9.30 
9.29 

Q  97 

9.989300 
.989271 
.989243 
.989214 
.989186 
.989157 
.989128 
.989100 

.47 

.47 
.47 
.48 

.48 
.48 
.48 

9.351697 
.352287 
.352876 
.353465 
.354053 
.354640 
.355227 
.355813 

9.84 
9.82 
9.81 
9.80 
9.79 
9.78 
9.76 

&7fi 

0.648303 
.647713 
.647124 
.646535 
.645947 
.645360 
.644773 
.644187 

20 
19 
18 
17 
16 
15 
14 
13 

48 

.345469 

y.&l 

.989071 

.48 

.356398 

./O 

.643602 

12 

49 

.346024 

9.26 
9.25 

.989042 

.48 
.48 

.356982 

9.74 
9.73 

.643018 

11 

60 
51 
52 
53 
54 
55 
56 

9.346579 
.347134 
.347687 
.348240 
.348792 
.349343 
.349893 

9.24 
9.22 
9.21 
9.20 
9.19 
9.17 

9.989014 

.988985 
.988956 
.988927 
.988898 
.988869 
.988840 

.48 
.48 

.48 
.48 
.48 
.48 

9.357566 
.358149 
.358731 
.359313 
.359893 
.360474 
.361053 

9.72 
9.70 
9.69 
9.68 
9.67 
9.66 

0.642434 
.641851 
.641269 
.640687 
.640107 
.639526 
.638947 

10 
9 
8 
7 
6 
5 
4 

57 

.350443 

9.16 

.988811 

.48 

.361632 

9.65 

.638368 

3 

68 
59 
6C 

.350992 
.351540 
.352088 

9.15 
9.14 
9.13 

.988782 
988753 

.988724 

.48 
.49 
.49 

.362210 
.362787 
.363364 

9.63 
9.62 
9.61 

.637790 
.637213 
.636636 

2 
I 
0 

M. 

Coelue. 

D.  1". 

Sine. 

D.  I". 

Cotung. 

D.I". 

Tang.   M 

103° 


770 


COSINES,    TANGENTS,    AND   COTANGENTS. 


253 


M. 

Sine. 

D.  1". 

Cosine. 

D.l«. 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 
4 

9.352038 
.352636 
.353181 
.35375$ 
.354271 

9.11 
9.10 
9.09 
9.03 
9  07 

9.988724 
.938695 
.988666 
.988636 
.988607 

.49 
.49 
.49 

.49 
49 

9.363364 
.363940 
.364515 
.365090 
.365664 

9.60 
9.59 
9.58 
9.57 

0.636636 
.636060 
.635485 
.634910 
.634336 

60 
59 
68 
67 
66 

6 
6 

r 

8 
9 

.354815 
.355353 
.355901 
.356443 
.356984 

9.'  05 
9.04 
9.03 
9.02 
9.01 

.93.3578 
.988548 
.988519 
.938489 
.938460 

149 
.49 
.49 
.49 
.49 

.366237 
.366810 
.367382 
.367953 
.368524 

9.55 
9.54 
9.53 
9.52 
9.51 
9.50 

.633763 
.633190 
.632618 
.632047 
.631476 

65 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.357524 

.358064 
.358603 
.359141 
.359678 
.360215 
.360762 
.361287 
.361822 
.362356 

8.99 
8.98 
8.97 
8.96 
8.95 
8.94 
8.92 
8.91 
8.90 
8.89 

9.988430 
.983401 
.988371 
.988342 
.988312 
.938282 
.988252 
.988223 
.988193 
.933163 

.49 
.49 
.49 
.60 
.60 
.60 
.60 
.60 
.60 
.60 

9.369094 
.369663 
•370232 
.370799 
.371367 
.371933 
.372499 
.373064 
.373629 
.374193 

9.49 
9.48 
9.47 
9.45 
9.44 
9.43 
9.42 
9.41 
9.40 
9.39 

0.630906 
.630337 
.829768 
.629201 
.628633 
.628067 
.627601 
.626936 
.626371 
.625807 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 

9.362889 
.363422 
.363954 
.364485 
.365016 
.365546 

8.88 
8.87 
8.86 
8.84 
8.83 

Q  OO 

9.988133 

.988103 
.938073 
.988043 
.988013 
.937983 

.60 
.50 
.60 
.60 
.50 
fin 

9.374756 
.375319 
.375881 
.376442 
.377003 
.377563 

9.38 
9.37 
9.36 
9.35 
9.33 

0.625244 
.624681 
.624119 
.623558 
.622997 
.622437 

40 
39 
38 
37 
36 
35 

26 
27 
23 
29 

.366075 
.366604 
.367131 
.367659 

O.O& 

8.81 
8.80 
8.79 
8.78 

937953 
.987922 
.987892 
987862 

.ou 
.50 
.50 
.60 
.51 

.378122 
.378681 
.379239 
.379797 

9.'31 
9.30 
9.29 
9.23 

621878 
.621319 
.620761 
.620203 

34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 

9.363185 
.368711 
369236 
369761 
.370285 
.370808 
.371330 
.371852 

8.76 

8.75 
8.74 
8.73 
8.72 
8.71 
8.70 
8  69 

9.987832 
.987801 
987771 
.987740 
.987710 
.987679 
.987649 
.987618 

.51 
.51 
.61 
.51 
.51 
.51 
61 

9.380354 
.380910 
.381466 
.382020 
.382575 
.383129 
.383682 
.384234 

9.27 
9.26 
9.25 
9.24 
9.23 
9.22 
9.21 

0.619646 
.619090 
.618534 
.617980 
.617425 
.616871 
.616318 
.615766 

30 
29 
28 
27 
26 
25 
24 
23 

38 

.372373 

.987538 

K1 

.384786 

Q  1  Q 

.615214 

22 

39 

.372894 

a  66 

.987557 

,ul 

.61 

.385337 

y.iy 
9.18 

.614663 

21 

40 
41 
42 
43 

9.373414 
.373933 
.374452 
.374970 

8.65 
8.64 
8.63 

Q  £Q 

9.987526 
.987496 
.987465 
.987434 

.61 
.61 
.51 

ei 

9.385888 

.386438 
.386987 
.387536 

9.17 
9.16 
9.15 

0.614112 
.613562 
.613013 
.612464 

20 
19 
18 
17 

44 
45 
i  46 

.375487 
.376003 
.376519 

O.O4 

8.61 
8.60 
o  cq 

.937403 
.987372 
.987341 

.Ol 

.51 
.52 

.388084 
.383631 
.389178 

9J2 
9.11 
Q  in 

.611916 
.611369 

.610822 

16 
15 
14 

47 

.377035 

o.oy 

Q  KQ 

.987310 

eo 

.389724 

y.  iu 

.610276 

13 

43 

.377549 

o.Oo 
o  c7 

.987279 

Mi 

.390270 

9.09 

.609730 

12 

49 

.378063 

O.O/ 

8.56 

.987248 

.52 
.52 

.390815 

9.08 
9.07 

.609185 

11 

60 
51 
52 
53 
54 
55 
56 
57 
68 
59 
60 

9.378577 
.379089 
.379601 
.330113 
.380624 
.331134 
.381643 
.382152 
.332661 
.333163 
.333676 

8.55 
8.53 
8.52 
8.51 
8.50 
8.49 
8.43 
8.47 
8.46 
8.45 

9.987217 
.937186 
.987155 
.987124 
.987092 
.987061 
.987030 
.936993 
.936967 
.986936 
.986904 

52 
.52 
.52 
.52 
.52 
.52 
.52 
52 
.52 
.52 

9.391360 
.391903 
.392447 
.392989 
.393531 
.394073 
.394614 
.395154 
.395694 
.396233 
.396771 

9.06 
9.05 
9.04 
9.03 
9.02 
9.01 
9.00 
8.99 
8.98 
8.97 

0.608640 
.608097 
.607553 
.607011 
.606469 
.605927 
.605386 
.604846 
.604306 
.603767 
.603229 

10 
9 
8 
7 
6 
6 
4 
3 

1 
0 

M. 

Cosine. 

D.I'. 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

MT 

1030 


TV* 


254 


TABLE   XV.       LOGATUTIIMIC    SINES, 


166* 


M 

Sine. 

D.  1". 

Coelne. 

D.I*. 

Tang. 

D.  1". 

Cotang 

M.  I 

0 

1 
2 
3 
4 
5 
6 
7 
8 

9.383675 
.384182 
.384687 
.385192 
.385697 
.386201 
.386704 
.387207 
.387709 

8.44 
8.43 
8.42 
8.41 
8.40 
8.39 
8.38 
8.37 

9.986904 
.986873 
.986841 
.986809 
.986778 
.986746 
.986714 
.986683 
.986651 

.53 
.63 
.53 
.53 
.53 
.63 
.53 
.53 

9.396771 
.397309 
.397846 
.398383 
.398919 
.399455 
399990 
.400524 
.401058 

8.96 
8.96 
8.95 
8.94 
8.93 
8.92 
8.91 
8.90 

Q  OQ 

0.603229 
.602691 
.602154 
.601617 
.601081 
.600545 
.600010 
.599476 
.598942 

so  ! 

59 
58 
57 
56 
55 
54 
53 
52 

9 

.388210 

8.36 
8.35 

.986619 

53 

.401591 

8.88 

.598409 

51 

10 
11 

9.388711 
.389211 

8.34 

9.986587 
986555 

.53 

9.402124 
.402656 

8.87 

0.597876 
.597344 

50 
49 

12 
13 
14 
15 
16 
17 
18 
19 

.389711 
.390210 
.390708 
.391206 
.391703 
.392199 
.392695 
.393191 

8.33 
8.32 
8.31 

8.30 
8.29 
8.28 
8.27 
8.26 
8.25 

.986523 
.986491 
.986459 
.986427 
986395 
.936363 
.986331 
.986299 

.63 
.53 
.53 
.63 
.54 
.54 
.54 
.54 
.54 

.403187 
.403718 
.404249 
.404778 
.405308 
.405836 
.406364 
.406892 

8.85 
8.84 
8.83 
8.82 
8.81 
8.80 
8.79 
8.78 

.296813 
.696282 
.595751 
.595222 
.594692 
.594164 
.593636 
.593108 

48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.393685 
.394179 
.394673 
.395166 
.395658 
.396150 
.396641 
.397132 
.397621 
.398111 

8.24 
8.23 
8.22 
8.21 
8.20 
8.19 
8.18 
8.17 
8.18 
8.15 

9.986266 
.936234 
.986202 
.986169 
.936137 
.986104 
.986072 
.986039 
.986007 
.985974 

.54 
.54 
.64 
.54 
.64 
.64 
.54 
.54 
.54 
.64 

9.407419 
.407945 
.408471 
.408996 
.409521 
.410045 
.410569 
.411092 
.411615 
.412137 

8.77 
8.76 
8.75 
8.75 
8.74 
8.73 
8.72 
8.71 
8.70 
8.69 

0.692581 
.592055 
.591529 
.591004 
.690479 
.589955 
.589431 
.588908 
.588385 
.587863 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

9.398600 

9.985942 

9.412658 

a  (*a 

0.687342 

30 

31 
32 
33 
34 
35 
36 
37 
38 

.399088 
.399575 
.400062 
.400549 
.401035 
.401520 
.402005 
.402489 

8.14 
8.13 
8.12 
8.11 
8.10 
8.09 
8.08 
8.07 

.985909 
.985876 
.985843 
.985811 
.985778 
.985745 
.985712 
.985679 

.54 
.55 
.55 
.65 
.55 
.55 
.65 
.55 

.413179 
.413699 
.414219 
.414738 
.415257 
.415775 
.416293 
.416810 

8.67 
8.66 
8.65 
8.65 
8.64 
8.63 
8.62 

Q  C1 

.586821 
.586301 
.585781 
.585262 
.684743 
.584225 
.583707 
.583190 

29 
28 
27 
26 
25 
24 
23 
22 

39 

.402972 

8.06 
8.05 

.985646 

.55 
.55 

.417326 

8.60 

.582674 

21 

40 
41 

42 
43 

9.403455 
.403938 
.404420 
.404901 

8.04 
8.03 
8.02 

9.985613 

.985580 
.985547 
.985514 

.55 
.55 
.55 

9.417842 
.418358 

.418873 
.419387 

8.59 

8.58 
8.57 

Q  EC 

0.582158 
.581642 
.581127 
.580613 

20 
19 
18 
17 

44 
45 

.405382 
.405862 

8.00 

.985480 
.985447 

.55 
.55 

.419901 
.420415 

8.56 

.580099 
.579585 

16 
15 

46 
47 

48 

.406341 
.406820 
407299 

7.99 
7.98 
7.97 

.985414 
.985381 
.985347 

.55 
.56 
.56 

.420927 
421440 
421952 

8.54 
8.53 

.579073 
.578560 
.578048 

14 
13 
12 

49 

.407777 

7.96 
7.96 

.985314 

.56 
56 

.422463 

8.51 

.577537 

11 

50 
51 

9.408254 
.408731 

7.95 

9.985280 
.985247 

.56 

9.422974 
.423484 

8.60 

0.577026 
.576516 

10 
9 

52 
53 

54 

.409207 
.409682 
.410157 

T.94 
7.93 
7.92 

.985213 

.985180 
.985146 

.56 
.56 
.56 

.423993 
.424503 
.425011 

8.49 
8.48 

.576007 
.575497 
.574989 

8 
7 
6 

55 
56 

.410632 
.411106 

7.91 
7.90 

.985113 

.985079 

.56 
.66 

.425519 
.426027 

8.46 

Q  AK 

.574481 
.573973 

5 
4 

57 

58 

.411579 
.412052 

7.88 

.985045 
.98501  1 

.56 
.56 

.426534 

.427041 

8.44 

,573466 
.572959 

3 
2 

59 
60 

.412524 
.412996 

7.87 
7.86 

.984978 
.984944 

.56 
.56 

.427547 
.428052 

8.43 

.572453 
.571948 

1 
0 

M. 

Cosine. 

D.I" 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Taug 

M. 

!| 

1040 


COSINES,    TANGENTS,    AND   COTANGENTS. 


255 
164 


M. 

Bine. 

D.P. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M 

0 

2 
3 
4 

5 
6 
7 
8 
9 

9.412996 
.413467 
.413938 
.414408 
414878 
.415347 
.415815 
.416283 
.416751 
.417217 

7.85 
7.84 
7.84 
7.83 
7.82 
7.81 
7.80 
7.79 
7.78 
7.77 

9.984944 
984910 
.984876 
.984842 
.984808 
.984774 
984740 
984706 
984672 
.984638 

56 
.57 
57 
67 
57 
.57 
.67 
.57 
.67 
.67 

9.428052 
.428558 
.429062 
.429566 
430070 
.430573 
.431075 
431577 
.432079 
432580 

8.42 
8.41 
8.40 
8.39 
8.38 
8.38 
8.37 
8.36 
8.35 
8.34 

0.571948 
571442 
570938 
570434 
569930 
569427 
-568925 
.568423 
567921 
.567420 

60 
59 
68 

57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 

9.417684 
.418150 
.4)3615 
419079 
.419544 
.420007 
420470 

7.76 
7.75 
7.75 
7.74 
7.73 
7.72 

7  71 

9.984603 
984569 
984535 
.984500 
984466 
.984432 
.984397 

67 

67 
.67 
.67 
67 
67 

CQ 

9.433080 
433580 
434080 
434579 
435078 
435576 
436073 

8.33 
8.33 
8.32 
8.31 
8.30 
8.29  . 

O  OQ. 

0.566920 
566420 
.565920 
.565421 
.564922 
.564424 
663927 

50 
49 
48 
4? 
46 
45 
44 

17 
18 

420933 
.421395 

7.70 

7  AQ 

.934363 
984328 

.68 

CO 

436570 
437067 

8.28 

Q  Q7 

663430 
.562933 

43 
42 

19 

421857 

7.68 

.984294 

.68 

437563 

8.26 

.662437 

41 

20 

9.422318 

7  R7 

9.984259 

CO 

9.433059 

U  OR 

0.661941 

40 

21 

22 

.422778 
*23238 

7.67 

.934224 
984190 

.68 

.438554 
.439048 

8.24 

.661446 
.560952 

39 
38 

23 
24 
25 
26 
27 
28 

423697 
.424156 
424615 
425073 
425530 
425987 

7.65 
7.64 
7.63 
7.62 
7.61 

984155 
.934120 
984085 
.984050 
.984015 
983981 

.68 
.68 
68 
68 

.68 

439543 

.440036 
440529 
441022 
441514 
442006 

8.23 
8.22 
8.21 
8.20 
8.20 

660457 
659964 
.659471 
.658978 

.658486 
.557994 

37 
36 
35 
34 
33 
32 

29 

426443 

7.61 
7.60 

.983946 

.58 
.68 

442497 

8.19 
8.18 

.557503 

31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.426899 
.427354 
427809 
428263 
428717 
.429170 
429623 
.430075 
430527 
430978 

7.59 
7.58 
7.67 
7.66 
7.65 
7.55 
7.53 
7.52 
7.62 
7.51 

9.983911 
.983875 
983840 
.983805 
983770 
983735 
983700 
983664 
933629 
983594 

.68 
68 
69 
59 
.69 
69 
69 
59 
69 
69 

9.442988 
443479 
443968 
444458 
444947 
445435 
445923 
446411 
446898 
447384 

8.17 
8.16 
8.16 
8.16 
8.14 
8.13 
8.13 
8.12 
8.11 
8.10 

0.657012 
.556521 
656032 
.555542 
655053 
.654565 
.654077 
.553589 
.553102 
.652616 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
13 
M 
45 
1  *6 

9.431429 
.431879 
.432329 
.432778 
.433226 
.433675 
434122 

7.50 
7.49 
7.49 
7.48 
7.47 
7.46 

9.983558 
933523 
933487 
983452 
983416 
983331 
983345 

59 
59 
69 
59 

59 
59 

9.447870 
448356 
448841 
449326 
449810 
450294 
450777 

8.09 
8.09 
8.08 
8.07 
8.06 
8.06 

0.552130 
651644 
651159 
.550674 
550190 
.549706 
.549223 

20 
19 
IS 
17 

16 
15 
14 

47 

.434569 

933309 

.59 

451260 

8.05 

.548740 

13 

43 
49 

435016 
435462 

7.44 
7.43 

983273 
983238 

.60 

.60 
60 

451743 
452225 

8.04 
8.03 
8.03 

.548257 
.547775 

12 
11 

60 

9.435908 

9.983202 

9.452706 

0.?  47294 

10 

51 

.436353 

933166 

.60 

.453187 

8.02 

.546813 

9 

52 

436793 

933130 

60 

453668 

546332 

8 

53 

.437242 

983094 

.60 

454148 

.545852 

7 

54 
55 

437636 
438129 

7.39 

983058 
983022 

.60 
,60 

.454628 
455107 

8.00 
7.99 

.545372 
.544893 

6 
5 

56 
57 
68 
59 
60 

438572 
439014 
.439456 
.439897 
.4403X3 

737 
7.36 
736 
7.35 

932986 
932950 
982914 
982878 
932342 

.60 
.60 
.60 
60 
60 

455586 
456064 
456542 
457019 
457496 

7.98 
7.97 
7.97 
7.96 
7.95 

.544414 
.543936 
.543458 
542981 
.542504 

4 

3 
2 
1 
0 

M. 

Cosine.  |  D.l". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

256           TABLE  XV.   LOGARITHMIC  SINES, 

160                                               163° 

M. 

Sine. 

D  1". 

Cosine. 

D.  i": 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 

4 
5 
6 

7 
8 
9 

9.440338 
.440778 
.441218 
.441658 
.442096 
.442535 
.442973 
.443410 
.443847 
.444234 

7.34 
7.33 
7.32 
7.31 
7.31 
7.30 
7.29 
7.28 
7.27 
727 

9.982842 

.982805 
.982769 

.932733 
.932696 
.932660 
.932624 
.932587 
.982551 
.932514 

.60 
.60 
.61 
.61 
61 
.61 
.61 
.61 
.61 
.61 

9.457496 
.457973 
.458449 
.458925 
.459400 
.459375 
.460349 
.460823 
.461297 
.461770 

7.94 
7.94 
7.93 
7.92 
7.91 
7.91 
7.90 
7.89 
7.83 
7.88 

0.542504 
.542027 
.541551 
.541075 
.540600 
.540125 
.539651 
.539177 
.638703 
.533230 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
H 
12 
13 
14 
15 
16 
17 
18 
19 

9.444720 
.445155 
.445590 
.446025 
.446459 
.446893 
.447326 
.447759 
.448191 
.448623 

7.26 
7.25 
7.24 
7.24 
7.23 
7.22 
7.21 
7.20 
7.20 
7  19 

9.982477 
.982441 
.982404 
.932367 
.982331 
.982294 
.982257 
.982220 
.982183 
.932146 

.61 
.61 
.61 
.61 
.61 
.61 
.61 
.62 
.62 
.62 

9.462242 

.462715 
.463186 
.463658 
.464128 
.464599 
.465069 
.465539 
.466008 
.466477 

7.87 
7.86 
7.86 
7.85 

7.84 
7.83 
7.83 
7.  82 
7.81 
7.81 

0.537758 
.537285 
.536814 
.536342 
.535372 
.535101 
.534931 
.534461 
.533992 
.533523 

60  ! 
49 

48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

9.449054 
.449485 
.449915 
450345 
.450775 
.451201 
.451632 
.452060 
.452488 
.452915 

7.18 
7.17 
7.17 
7.16 
7.15 
7.14 
7.13 
7.13 
7.12 
7.11 

9.932109 
.982072 
.932035 
.981993 
.981961 
.931924 
.931836 
.931849 
.931812 
.981774 

.62 
.62 
.62 
62 
.62 
.62 
.62 
.62 
.62 
62 

9.466945 
.467413 
.467880 
.468347 
.463814 
.469280 
.469746 
.470211 
.470676 
.471141 

7.80 
7.79 
7.78 
7.78 
7.77 
7.76 
7.76 
7.76 
7.74 
7.74 

0.533055 
.532587 
532120 
.531653 
.531186 
.530720 
.530254 
.529789 
.529324 
.528859 

40  , 
39 
38 
37 
36 
36 
34 
33  ( 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.453342 
.453768 
.454194 
.454619 
.455044 
.455469 
.455893 
.456316 
.456739 
.457162 

7.10 
7.10 
7.09 
7.08 
7.07 
7.07 
7.06 
7.05 
7.04 
704 

9.981737 

.981700 
.931662 
.981625 
.981587 
.931549 
.981512 
.981474 
.931436 
.931399 

.62 
.62 
.63 
.63 
.63 
.63 
.63 
.63 
.63 
.63 

9.471605 
.472069 
.472532 
.472995 
.473457 
.473919 
.474381 
.474842 
.475303 
.475763 

7.73 
7.72 
7.71 
7.71 
7.70 
7.69 
7.69 
7.68 
7.67 
7.67 

0.528395 
.627931 
.527468 
.527005 
.526543 
.526081 
.525619 
.525158 
.524697 
.624237 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.457584 
.458006 
.458427 
.458848 
.459268 
.459688 
.460108 
.460527 
.460946 
.461364 

7.03 
7.02 
7.01 
7.01 
7.00 
6.99 
6.98 
6.98 
6.97 
696 

9.981361 
.981323 
.981285 
.981247 
.981209 
.981171 
.981133 
.981095 
.981057 
.981019 

.63 
.63 
.63 
.63 
.63 
.63 
.63 
.64 
.64 
.64 

9.476223 
.476683 
.477142 
.477601 
.478059 
.478517 
.478975 
.479432 
.479889 
.480345 

7.66 
7.65 
7.65 
7.64 
7.63 
7.63 
7.62 
7.61 
7.61 
7.60 

0.523777 
.523317 
.522858 
.522399 
.521941 
.521483 
.521025 
.520568 
.520111 
.519655 

20 
19 
18 

16 
16 
14 
13 
12 
11 

50 
61 
52 

9.461782 
.462199 
.462616 

6.96 
6.95 

9.980931 
.980942 
.980904 

.64 
.64 

9.480801 
.481257 
.481712 

7.59 
7.59 

0.519199 
.518743 
.518288 

10 
9 

8 

63 
64 
55 
56 

67 
58 
59 
60 

.463032 
.463448 
.463364 
.464279 
.464694 
.465108 
.465522 
.465935 

6.94 
6.93 
6.93 
6.92 
6.91 
6.90 
6.90 
6.89 

.980366 
.980827 
.980789 
.980750 
.980712 
.980673 
.980635 
.980596 

.64 
.64 
.64 
.64 
.64 
.64 
.64 
.64 

.482167 
.482621 
.483075 
.483529 
.483982 
.484435 
.484887 
.485339 

7.57 
7.57 
7.56 
7.55 
7.55 
7.54 
7.53 

.517833 
.617379 
.516925 
.516471 
.516018 
.515565 
.515113 
.514661 

7 
6 
5 
4 
3 
2 

0 

M. 

Cosine. 

D.  1". 

Shuv 

-ZZ5Z 

D.l". 

Cotang. 

D.  1". 

Pang. 

M. 

COSINES,  TANGENTS,  AND  COTANGENTS.        25  "J 

17°                                               16JT 

M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

9.465935 

fi  QQ 

9.930596 

64 

9.485339 

f  £Q 

0.514661 

60 

1 

.466343 

O.OO 
A  ftQ 

.930558 

.435791 

/  .Oo 

.514209 

59 

2 
3 
4 
6 
6 
7 
8 
9 

.466761 
.467173 
.467585 
.467996 
.468407 
.468817 
.469227 
.469637 

O.OO 

6.87 
6.86 
6.85 
6.85 
6.84 
6.83 
6.83 
6.82 

980519 
.980480 
980442 
.980403 
930364 
980325 
930236 
.980247 

.65 
.65 
.65 
.65 
.65 
.65 
.65 
.65 

.436242 
.486693 
.487143 
.487593 
.488043 
.488492 
.488941 
.489390 

7.52 
7.51 
7.51 
7.50 
7.50 
7.49 
7.48 
7.48 
7.47 

.513758 
.513307 
.512857 
.512407 
.511957 
.511508 
.511059 
.510610 

58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 

9.470046 
.470455 
.470863 
.471271 
.471679 

6.81 
6.81 

6.80 
6.79 

9.980208 
.980169 
980130 
.980091 
.980052 

.65 
.65 
.65 
.65 

9.489838 
.490286 
.490733 
.491180 
.491627 

7.46 
7.46 
7.45 
7.44 

0.510162 
.509714 
.509267 
.508820 
.508373 

50 
49 

48 
47 
46 

15 
16 
17 
18 

.472086 
.472492 
.472898 
.473304 

6.78 
6.78 
6.77 
6.76 

.980012 
979973 
.979934 
.979895 

.65 
.65 
.65 
.66 

.492073 
.492519 
.492965 
.493410 

7.44 
7.43 
7.43 
7.42 

.507927 
.507481 
.507035 
.506590 

45 
44 
43 
42 

19 

473710 

6.76 
6.75 

.979855 

.66 
.66 

.493354 

7.41 
7.41 

.506146 

41 

20 
21 

9.474115 
.474519 

6.74 

9.979816 
.979776 

.66 

9.494299 
.494743 

7.40 

0.505701 
.505257 

40 
39 

22 
23 

.474923 
.475327 

6.74 
6.73 

.979737 
.979697 

.66 
.66 

.495186 
.495630 

7.39 
7.39 

.604814 
.604370 

33 
37 

24 

.475730 

6.72 

.979653 

.66 

.496073 

7.38 

.503927 

36 

25 
26 

.476133 
.476536 

6.72 
6.71 

.979618 
.979579 

.66 
.66 

.496515 

.496957 

7.38 
7.37 

.503485 
.503043 

35 
34 

27 

.476938 

6.70 

.979539 

.66 

497399 

7.36 

.502601 

33 

28 

.477340 

6.69 

.979499 

.66 

.497841 

7.36 

.502159 

32 

29 

.477741 

6.69 
6.63 

.979459 

.66 
.66 

.498282 

7.35 
7.34 

.501718 

31 

30 

9.478142 

9.979420 

9.498722 

0.501278 

3(1 

31 

.478542 

6.67 

.979380 

66 

.499163 

7.34 

.500837 

29 

32 

.478942 

6.67 

.979340 

.66 

.499603 

7.33 

.500397 

28 

33 
34 
35 

.479342 
.479741 
.480140 

6.66 
6.65 
6.65 

.979300 
.979260 
.979220 

.67 
.67 
.67 

.500042 
.500431 
.500920 

7.33 
7.32 
7.31 

.499958 
.499519 
.499080 

27 
26 
25 

36 
37 

.480539 
.480937 

6.64 
6.63 

.979180 
.979140 

.67 
.67 

.501359 
.501797 

7.31 
7.30 

.498641 
.498203 

24 
23 

38 
39 

.481334 
.481731 

6.63 
6.62 

.979100 
.979059 

.67 
.67 

.502235 
502672 

7.30 
7.29 

.497765 
.497328 

22 
21 

6.61 

.67 

7.28 

40 
41 
42 

9.482128 
.482525 
.482921 

6.61 
6.60 

9.979019 
.978979 
.978939 

.67 
.67 

9.503109 
503546 
503932 

7.28 
7.27 

0.496891 
.496454 
.496018 

20 
19 

18 

43 
44 

.483316 
.483712 

6.59 
6.59 

.978393 
.978858 

.67 

.67 

.504418 

.504354 

7.27 
7.26 

.495582 
.495146 

17 
16  ' 

45 
46 

.484107 
.434501 

6.58 
6.57 

.978817 
.978777 

.67 
.67 

.505239 
.505724 

7.25 
7.25 

.494711 
.494276 

15 
14 

47 

.434895 

6.57 

.978737 

.67 

.506159 

7.24 

.493841 

13 

48 

.485289 

6.56 

.978696 

.68 

.506593 

7.24 

.493407 

12 

49 

.485682 

6.55 

.978655 

.68 

.507027 

7.23 

.492973 

11 

6.55 

68 

7.23 

50 

9.486075 

9.978615 

9.507460 

0.492540 

10  i 

51 

.486467 

6.54 

.978574 

.68 

.507893 

7.22 

.492107 

9 

52 
53 
54 

.486360 
.487251 
.437643 

6.54 
6.53 
6.52 

.978533 
.978493 
.978452 

.63 
.68 
68 

.503326 
.508759 
.509191 

7.21 
7.21 
7.20 

.491674 
.491241 
.490809 

8 
7 
6 

55 

.438034 

6.52 

.978411 

.68 

.509622 

7.20 

.490378 

5 

56 

.483424 

6.51 

.978370 

.68 

.510054 

7.19 

.489946 

4 

57 

58 

.483814 
.489204 

6.50 
6.50 

.978329 

.978238 

.68 
.68 

.510435 
.510916 

MS 

.489515 
.489084 

3 
2 

59 

.439593  1   °  ™ 

.978247 

.63 

511346 

7.17 

.488654 

1 

60 

.489932  !   6'48 

.973206 

.68 

.511776 

7.17 

.438224 

0 

M. 

Cosine.   D.  1". 

Sloe. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

18 


73° 


258           TABLE  XV.   LOGARITHMIC  SINES, 

180                                               1610 

M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

~T 

9.489982 

A  4ft 

9.978206 

AQ 

9.511776 

0.488224 

60 

2 
3 

.490371 
.490759 
.491147 

O.4O 

ti.47 
6.46 

.978165 
.978124 
.978083 

,Oo 

.69 
.69 

.512206 
.512635 
.513064 

7.16 
7.16 
7.15 

.487794 
.487365 
486936 

59 
63 
67 

4 

.491535 

2*52 

.978042 

.69 

513493 

186507 

66 

5 

.491922 

6.45 

.978001 

.69 

.513921 

l-\\  ,   486079 

66 

6 

.492308 

A  44 

.977959 

an 

.514349 

»»  10 

.485651 

64 

7 
8 
9 

.492695 
.493081 
.493466 

BA3 
6.43 
6.42 

.977918 
.977877 
.977835 

.69 
.69 
.69 
.69 

.514777 
.615204 
.515631 

7.13 
7.12 
7.12 
7.11 

485223 
.484796 
.484369 

63 
62 
61 

10 

9.493851 

9.977794 

9.516057 

0.483943 

60 

11 

.494236 

o.41 

A  41 

.977752 

.69 

AQ 

.616484 

7.10 

.483516 

49 

12 

.494621 

O.41 
A  4H 

.977711 

.oy 

.616910 

7.10 

.483090 

48 

13 

.495005 

O.4U 

.977669 

*2J 

517335 

7.09 

.482665 

47 

14 

.495388 

6.39 

A  QQ 

.977628 

69 

.617761 

7.09 

.482239 

46 

15 
16 
17 
18 
19 

.495772 
.496154 
.496537 
.496J19 
.497301 

o.oy 
6.33 
6.38 
6.37 
6.36 
6.36 

.977586 
.977544 
.977503 
.977461 
.977419 

.69 
.70 
.70 
.70 
.70 

.518186 
.518610 
.519034 
.519458 
.519882 

7.08 
7.08 
7.07 
7.07 
7.06 
7.05 

.481814 
.481390 
.480966 
.480542 
.480118 

45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 

9.497682 

.498064 
.498444 
.498825 
.499204 
.499584 
.499963 
.600342 
.500721 

6.35 
6.34 
6.34 
6.33 
6.33 
6.32 
6.31 
6.31 
A  on 

9.977377 
.977335 
.977293 
.977251 
.977209 
.977167 
.977125 
.977083 
.977041 

.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 

9.520305 
.620728 
.521151 
.621673 
.521995 
.622417 
.522838 
.523259 
.523680 

7.05 
7.04 
7.04 
7.03 
7.03 
7.02 
7.02 
7.01 

0.479695 
.479272 
.478849 
.478427 
.478005 
.477583 
.477162 
.476741 
.476320 

40 
39 
38 
37 
36 
35 
34 
33 
32 

29 

.501099 

O.OU 

6.30 

976999 

.70 
.70 

.524100 

r.oi 

7.00 

.476900 

31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.501476 
.501854 
.502231 
.502607 
.502984 
.503360 
.503735 
.604110 
.504485 
.604860 

6.29 
6.28 
6.28 
6.27 
6.27 
6.26 
6.25 
6.25 
6.24 
6.24 

9.976957 
.976914 
.976872 
.976830 
.976787 
.976745 
.976702 
.976660 
976617 
.976574 

.70 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 

9.524520 
.524940 
.525359 
.625778 
626197 
.626615 
.627033 
.627451 
.627868 
.528285 

6.99 
6.99 
6.98 
6.98 
6.97 
6.97 
6.96 
6.96 
6.95 
6.95 

0.476480 
.476060 
.474641 
.474222 
.473803 
.473385 
472967 
.472549 
.472132 
.471716 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.605234 
.605608 
.505981 
.506354 
.606727 
.607099 
.607471 
.607843 
.508214 
.508585 

6.23 
6.22 
6.22 
6.21 
6.21 
6.20 
6.19 
6.19 
6.18 
6.18 

9.976532 
.976489 
.976446 
.976404 
.976361 
.976318 
.976275 
.976232 
.976189 
.976146 

.71 
.71 
.71 
.71 
.71 
.72 
.72 
.72 
.72 
.72 

9.528702 
.529119 
.629535 
.629951 
.530366 
.530781 
.531196 
.531611 
.532025 
.532439 

6.94 
6.94 
6.93 
6.93 
6.92 
6.91 
6.91 
6.90 
6.90 
6.89 

0.471298 
.470881 
.470465 
.470049 
.469634 
.469219 
.468804 
.468389 
467975 
.467561 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
51 
52 
53 
64 
55 
66 
57 
53 

9.503956 
.509326 
.50%96 
.510065 
.510434 
.610803 
.M1172 
.511540 
.511907 

6.17 
6.16 
6.16 
6.15 
6.15 
6.14 
6.14 
6.13 

9.976103 
.976060 
.976017 
.975974 
.975930 
.975887 
.975844 
.975800 
.975757 

.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 

9.532853 
.533266 
.533679 
.534092 
.534504 
.534916 
.535328 
.535739 
.536150 

6.89 
6.88 
6.88 
6.87 
6.87 
6.86 
6.86 
6.85 

0.467147 
.466734 
.466321 
465908 
.465496 
.465084 
.464672 
464261 
.463850 

10 
9 
8 
7 
6 
5 
4 
3 
2 

69 

.512275 

6.12 

.975714 

.72 

536561 

6.85 

.463439 

1 

60 

.512642 

6.12 

.975670 

.72 

.536972 

6.84 

.463028 

0 

If. 

Co&iuo. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

108" 


190 


COSINES,    TANGENTS,    AND   COTANGENTS. 


M 

Sine. 

D.F. 

Cofilne. 

D.  1". 

Tang. 

D.  1". 

Cotaag. 

M. 

0 
1 

9.512642 
.513009 

6.11 

All 

9.975670 

.975627 

.73 
73 

9.536972 
.537382 

6.84 

0.463028 
.462618 

60 
59 

2 
3 
4 
5 

6 

7 
8 
9 

.513375 
.513741 
.514107 
.614472 
.514837 
.515202 
515566 
515930 

6.10 
6.09 
6.09 
6.08 
6.08 
6.07 
6.07 
6.06 

.975583 
.975589 
.975496 
.975452 
.975403 
.975365 
.975321 
.975277 

.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 

.537792 
.538202 
.533611 
.539020 
.539429 
.539337 
.540245 
.540653 

6.33 
6.82 
6.82 
6.81 
6.81 
6.80 
6.80 
6.79 

.462208 
.461798 
.461389 
.460980 
.460571 
.460163 
.459755 
.459347 

58 
57 
56 
65 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.516294 
.516657 
.517020 
.517382 
.517745 
.518107 
.518468 
.518829 
.519190 
.519551 

6.05 
6.05 
6.04 
6.04 
6.03 
6.03 
6.02 
6.02 
6.01 
6.00 

9.975233 
.975189 
.975145 
.975101 
.975057 
.975013 
.974969 
.974925 
.974880 
.974836 

.73 
.73 
.73 
.73 
.73 
.74 
.74 
.74 
.74 
.74 

9.541061 
.541463 
.541875 
.542281 
.542688 
.543094 
.543499 
.543905 
.644310 
.644715 

6.79 
6.78 
6.78 
6.77 
6.77 
6.76 
6.76 
6.75 
6.75 
6.74 

0.458939 
.458532 
.458125 
.457719 
.457312 
.456906 
.456501 
.456095 
.455690 
.455285 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.519911 
.520271 
.520631 
.520990 
.521349 
.521707 
.522066 
.522424 
.522781 
.523133 

6.00 
5.99 
5.99 
5.98 
5.93 
6.97 
5.97 
5.96 
5.95 
5.95 

9.974792 
.974748 
.974703 
.974659 
.974614 
.974570 
.974525 
.974481 
.974436 
.974391 

.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.75 

9.545119 
.545524 
.545928 
.546331 
.546735 
.547138 
.547540 
.547943 
.548345 
.548747 

6.74 
6.73 
6.73 
6.72 
6.72 
6.71 
6.71 
6.70 
6.70 
6.69 

0.454881 
.454476 
.454072 
.453669 
.453265 
.452862 
.452460 
.452057 
.451655 
.451253 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 

9.523495 
.523852 
.524208 

5.94 
5.94 

K  QQ 

9.974347 
.974302 
.974257 

.76 

.75 

9.549149 
.549550 
.549951 

6.69 
6.68 

0.450351 
.450450 
.450049 

30 
29 
23 

33 
34 
35 
36 
37 

.524564 
.524920 
.525275 
.525630 
.525984 

593 
6.92 
5.92 
5.91 

C  Qf) 

.974212 
.974167 
.974122 
.974077 
.974032 

.75 
.75 
.75 
.75 

.550352 
.550752 
.551153 
.551552 
.551952 

6.67 
6.67 
6.67 
6.66 

.449648 
.449248 

.448847 
.448448 
.448048 

27 
26 
25 
24 
23 

38 
39 

.526339 
.526693 

5.90 
5.89 

.973987 
.973942 

.75 

.76 

.552351 
.552750 

6.65 
6.65 

.447649 
.447250 

22 
21 

40 
41 
42 
43 

44 
45 
46 
47 

48 
49 

9.527046 
.527400 
.527753 
.528105 
.528458 
.528810 
.529161 
.529513 
.529364 
.530215 

5.89 
5.88 
5.88 
6.87 
5.87 
5.86 
5.86 
5.85 
5.85 
5.84 

9.973897 
.973852 
.973807 
.973761 
.973716 
.973671 
.973625 
.973580 
.973535 
.973439 

.75 

.75 
.75 
.75 
.76 
.76 
.76 
.76 
.76 
.76 

9.553149 
.553548 
.553946 
.554344 
.554741 
.555139 
.555536 
.555933 
.556329 
.556725 

6.64 
6.64 
6.63 
6.63 
6.62 
6.62 
6.61 
6.61 
6.60 
660 

0.446851 
.446452 
.446054 
.445656 
.445259 
.444361 
.444464 
.444067 
.443671 
.443275 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

1  50 
51 
52 

9.530565 
.530915 
.531265 

5.83 
5.53 

C  oo 

9.973444 
.973398 
.973352 

.76 
.76 

9.557121 
.557517 
.557913 

6.59 
6.59 

0.442879 
.442483 

.442037 

10 
9 
8 

53 
54 
55 

.531614 
.531963 
.532312 

5.82 
5.81 

C  Ol 

.973307 
.973261 
.973215 

.76 
.76 

.558308 

.558703 
.559097 

6.59 
6.53 
6.58 

.441692 
.441297 
.440903 

7 
6 
5 

56 
57 
58 
59 
60 

.532661 
.533009 
.533357 
.533704 
.534052 

5.80 
5.80 
5.79 
5.79 

.973169 
.973124 
.973078 
.973032 
.972986 

.76 

.76 
.76 
.77 
.77 

.559491 
.559385 
.560279 
.560673 
.561066 

6.57 
6.57 
6.56 
6.56 
6.55 

.440509 
.440115 
.439721 
.439327 
.438934 

4 
3 
2 
1 

0 

M. 

Oorine. 

D.  1". 

Slue. 

D.  1". 

Cotuug. 

D.  F. 

Tang. 

M. 

TABLE   XV.       LOGARITHMIC    SINES, 


169* 


M. 

Sine. 

D.  1". 

Coeine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M, 

0 
1 
2 
3 
4 

9.534052 
.534399 
.534745 
.535092 
.535438 

6.78 
5.78 
5.77 
5.77 

9.972986 
.972940 
.972894 

.972848 
.972802 

.77 
.77 
.77 
.77 

77 

9.561066 
.561459 
.561851 
.562244 
.562636 

6.55 
6.54 
6.54 
6.54 
6  53 

0.438934 
.438541 
.438149 
.437756 
.437364 

60 

59 
58 
57 
56 

5 
6 

7 
8 
9 

.535783 
.536129 
.536474 
.536818 
.637163 

5.76 
5.75 
6.75 
5.74 
5.74 

.972755 
.972709 
.972663 
.972617 
.972570 

.77 
.77 
.77 
.77 
.77 

.563028 
.563419 
.563811 
.564202 
.564593 

6.53 
6.52 
6.52 
6.51 
6.51 

.436972 
.436581 
.436189 
.435798 
.435407 

55 
54 
53 
52 
61 

10 
11 
12 
13 
14 
15 
16 
17 
18 

9.537507 
.537851 
.538194 
.538538 
.538880 
.539223 
.539565 
539907 
.540249 

5.73 
5.73 
6.72 
5.71 
5.71 
5.70 
5.70 
5.69 

9.972524 
.972478 
.972431 
.972385 
.972338 
.972291 
.972245 
.972198 
.972151 

.77 
.77 
.78 
.78 
.78 
.78 
.78 
.78 

fO 

9.564983 
.565373 
.565763 
.566153 
.566542 
.666932 
.567320 
.567709 
.568098 

6.50 
6.50 
6.50 
6.49 
6.49 
6.48 
6.48 
6.47 

fi  47 

0.435017 
.434627 
.434237 
.433847 
.433458 
.433068 
.432680 
.432291 
.431902 

50 
49 
48 
47 
46 
45 
44 
43 
42 

19 

.540590 

6.68 

.972105 

.78 

.568486 

6.46 

.431514 

41 

20 
21 
22 
23 
24 
25 

9.540931 
.541272 
.541613 
.541953 
.542293 
.542632 

5.68 
5.67 
5.67 
5.66 
5.66 

9.972058 
.972011 
.971964 
.971917 
.971870 
.971823 

.78 
.78 
.78 
.78 
.78 

7ft 

9.668873 
.569261 
.569648 
.670035 
.570422 
.670809 

6.46 
6.46 
6.45 
6.45 
0.44 

ft  44 

0.431127 
.430739 
.430352 
.429965 
.429578 
.429191 

40 
39 
38 
37 
36 
35 

26 
27 

28 
29 

.542971 
.543310 
.543649 
.643987 

6.65 
5.64 
6.64 
5.63 

.971776 
.971729 
.971682 
.971635 

.78 
.79 
.79 
.79 

.671195 
.671581 
.571967 
.572352 

6.43 
6.43 
6.43 
6.42 

.428805 
.428419 
.428033 
.427648 

34 
33 
32 

31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.644325 
.544663 
.645000 
645338 
.545674 
.646011 
646347 
.546683 
.647019 
.647354 

5.63 
6.62 
6.62 
5.61 
6.61 
6.60 
6.60 
6.59 
6.69 
6.58 

9.971588 
.971540 
.971493 
.971446 
.971398 
.971351 
.971303 
.971256 
.971208 
.971161 

.79 
.79 
.79 
.79 
.79 
.79 
.79 
.79 
.79 
.79 

9.572738 
.573123 
.673507 
.673892 
.674276 
.574660 
.676044 
.675427 
.675810 
.676193 

6.42 
6.41 
6.41 
6.40 
6.40 
6.40 
6.39 
6.39 
6.38 
6.38 

0.427262 
.426877 
.426493 
.426108 
.425724 
.425340 
.424956 
.424573 
.424190 
.423807 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.647689 
.648024 
.548359 
.648693 
.549027 
.549360 
.549693 
.550026 
.550359 
.550692 

5.58 
5.67 
6.57 
5.56 
6.56 
6.55 
5.55 
6.55 
6.64 
664 

9.971113 
.971066 
.971018 
.970970 
.970922 
.970874 
,970827 
.970779 
.970731 
.970683 

.79 

.80 
.80 
.80 
.80 
.80 
.80 
.80 
.80 
.80 

9.576576 
.576959 
.577341 
.577723 
.578104 
.578486 
.578867 
.579248 
.679629 
.580009 

6.37 
6.37 
6.37 
6.36 
6.36 
6.35 
6.35 
6.34 
6.34 
6.34 

0.423424 

.423041 
.422659 
.422277 
.421896 
.421614 
.421133 
.420762 
.420371 
.419991 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
68 
59 
60 

9.551024 
.651356 
.551687 
.552018 
.552349 
.552680 
.553010 
.653341 
.653670 
.654000 
.554329 

6.53 
5.53 
6.52 
5.52 
5.51 
5.51 
5.50 
6.50 
6.49 
6.49 

9.970635 
.970586 
.970538 
.970490 
.970442 
.970394 
.970345 
.970297 
.970249 
.970200 
.970152 

.80 
.80 
.80 
.80 
.80 
.81 
.81 
.81 
.81 
.81 

9.580389 
.580769 
.681149 
.581528 
.681907 
.582286 
.582665 
.683044 
.683422 
.583800 
.584177 

6.33 
6.33 
6.32 
6.32 
6.32 
6.31 
6.31 
6.30 
6.30 
6.30 

0.419611 
.419231 
.418851 
.418472 
.418093 
.417714 
.417335 
.416956 
.416578 
.416200 
.415823 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0 

M. 

Otelue. 

D.  1'  . 

Biiio. 

D.  1". 

Ootaug. 

D.  1". 

Tang. 

M. 

1100 


C9° 


ate 


COSINES,    TANGENTS,    AND    COTANGENTS. 


M. 

Sloe. 

D.I". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 

3 
4 
5 
6 

7 
8 
9 

9.554329 
.554658 
.554987 
.655315 
.555643 
.555971 
.556299 
.556626 
556953 
.557280 

6.48 
5.48 
6.47 
6.47 
6.46 
6.46 
6.45 
6.45 
5.44 
5.44 

9.970152 
.970103 
.970055 
.970006 
.969957 
.969909 
.&69SGO 
.969811 
.969762 
.969714 

.81 
.81 
.81 
.81 
81 
.81 
.81 
.81 
.81 
.81 

9.584177 
.584555 
.584932 
.585309 
.685686 
.586062 
.586439 
.586815 
.687190 
.687566 

6.29 
6.29 
6.28 
6.28 
6.28 
6.27 
6.27 
6.26 
6.26 
6.26 

0.415823 
.415445 
.415068 
.414691 
.414314 
.413938 
.413561 
.413185 
.412810 
.412434 

60 
59 
68 
67 
66 
55 
64 
63 
52 
61 

10 
11 

9  557606 
.557932 

6.44 

9.969665 
.969616 

.82 

9.587941 
.588316 

6.25 

0.412059 
.411684 

60 
49 

12 

.558258 

5.43 

.969567 

.82 

QQ 

588691 

6.25 

.411309 

48 

13 

.558583 

5.43 

.969518 

.CM 

.589066 

6.24 

.410934 

47 

14 
15 
16 
17 

.558909 
.559234 
.559558 
.559883 

5.42 
5.42 
5.41 
6.41 

.969469 
.969420 
.969370 
.969321 

.82 
62 
.82 
.82 

QQ 

.589440 
589814 
590188 
.690562 

6.24 
6.24 
6.23 
6.23 

.410560 
.410186 
.409812 
.409438 

46 
45 
44 
43 

18 

.560207 

6.40 

.969272 

.Ko 

.590935 

6.22 

.409065 

42 

19 

.560531 

6.40 
6.39 

.969223 

.82 
.82 

.591308 

6.22 
6.22 

.408692 

41 

20 
21 
22 
23 

9.560855 
.561178 
.661501 
.561824 

5.39 
6.38 
6.38 

9.969173 
969124 
.969075 

.969025 

.82 
.82 
'  .82 

oo 

9.591681 
.592054 
.592426 
.692799 

6.21 
6.21 
6.20 

0.408319 
.407946 
.407574 
.407201 

40 
39 
38 
37 

24 
25 
26 
27 

28 

.562146 
.562468 
.562790 
.563112 
.663433 

6.37 
6.37 
6.37 
5.36 
6.36 
c  qc 

.968976 
.968926 
.968877 
.968827 
.968777 

.0.6 

.83 
.83 
.83 
.83 

oq 

.593171 
.593542 
.593914 
.694285 
.594656 

6.20 
6.20 
6.19 
6.19 
6.18 

.406829 
.406458 
.406086 
.405715 
.405344 

36 
35 
34 
33 
32 

29 

.563755 

o.oo 

.968728 

.OO 

.695027 

6.18 

.404973 

31 

5.36 

.83 

6.18 

30 

9.564075 

K  1A 

9.968678 

oq 

9.595398 

0.404602 

30 

31 
32 
33 

.664396 
.564716 
.565036 

O.«H 

5.34 
5.33 

.968628 
.968678 
.968528 

.00 

83 
.83 

.595768 
.596138 
.596508 

6.17 
6.17 
6.16 

.404232 
.403862 
.403492 

29 
28 
27 

34 
35 

.565366 
.565676 

5.33 
5.32 

.968479 
.968429 

.83 
.83 

.596878 
.597247 

6.16 
6.16 

.403122 
.402753 

26 
25 

36 
37 
38 
39 

.565995 
.566314 
.566632 
.566951 

5.32 
5.32 
6.31 
5.31 
6.30 

.968379 
.968329 
.968278 
.968228 

.83 
.83 
.83 

.84 
.84 

.597616 
.597985 
.598354 
.598722 

6.15 
6.15 
6.15 
6.14 
6.14 

.402384 
.402015 
.401646 
.401278 

24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 

9.567269 
.567587 
.567904 
.568222 
.568539 
.568856 
.569172 
.569488 

6.30 
6.29 
5.29 
6.28 
6.28 
6.28 
6.27 

9.968178 
.968128 
.968078 
968027 
.967977 
.967927 
.967876 
.967826 

84 
.84 
.84 
.84 
.84 
.84 
.84 

9.599091 
.599459 
.599.827 
.600194 
.600562 
.600929 
.601296 
.601663 

6.13 
6  13 
6.13 
6.12 
6.12 
6.12 
6.11 

0.400909 
.400541 
.400173 
.399806 
.399438 
.399071 
.398704 
.398337 

20 
19 
18 
17 
16 
15 
14 
13 

48 
49 

.569804 
.670120 

5.27 
6.26 
5.26 

.967775 
.967725 

.84 

.84 
.84 

.602029 
.602395 

6.11 
6.10 
6.10 

.397971 
.397605 

12 
11 

50 
51 
52 
53 
54 
55 
66 

9.570435 
.570751 
.571066 
.571380 
.671695 
.572009 
.572323 

6.25 
6.25 
6.24 
624 
6.24 
6.23 

9.967674 
.967624 
.967573 
.967522 
.967471 
.967421 
.967370 

.84 
.84 
.85 
.85 
.85 
.85 

9.602761 
.603127 
.603493 

.603858 
.604223 
.604588 
.604953 

6.10 
6.09 
6.09 
6.09 
6.08 
6.08 

0.397239 
.396873 
.396507 
.396142 
.395777 
.395412 
.395047 

10 
9 
8 
7 
6 
5 
4 

67 

53 

.672636 
672950 

6.23 
6.22 

.967319 
.967268 

.85 

.85 

.605317 
.605682 

6.07 
6.07 

.394683 
.394318 

3 
2 

69 
60 

.673263 
.573575 

6.22 
6.21 

.967217 
.967166 

.85 

.85 

.606046 
.606410 

6.07 
6.06 

.393954 
.393590 

1 
0 

M. 

OOBIUO. 

D.  1". 

Sine. 

D.  1". 

Octai*. 

D.  1". 

Tang. 

M. 

Llio 


68° 


262 


TABLE  XV.      LOGARITHMIC   SINES, 


M. 

Sine. 

D.  1".   Oodne. 

D.  1". 

Tang. 

D.  1". 

Co  tang. 

M. 

0 
1 

9.573575 
.673888 

6.21 

9.967166 
.967115 

.85 

oe 

9.606410 
.606773 

6.06 

ft  Oft 

0.393590 
.393227 

60 
59 

2 
3 
4 

.674200 
.674512 
.574824 

6.2C 
5.2C 
6.20 

.967064 
.967013 
.966961 

.OO 

.85 
.85 

.607137 
.607500 
.607863 

o.uo 
6.05 
6.05 

.392863 
.392500 
.392137 

58 
57 
56 

5 
6 

.575136 
.575447 

5.19 
5.19 

.966910 
.966859 

.85 
.85 

.608225 

.608588 

e'.oi 

.391775 
.391412 

55 
54 

7 

.575758 

5.18 

.966808 

.86 

.608950 

6.04 

.391050 

53 

8 

.676069 

6.18 

.966756 

.86 

.609312 

6.03 

.390688 

52 

9 

.576379 

5.17 
6.17 

.966705 

.86 
.86 

.609674 

6.03 
6.03 

.390326 

51 

10 
11 
12 
13 
14 

9.676689 
.576999 
.577309 
.577618 
.677927 

6.17 
5.16 
5.16 
5.15 

9.966653 
.966602 
.966550 
.966499 
.966447 

.86 
.86 
.86 

.86 

9.610036 
.610397 
.610759 
.611120 
.611480 

6.02 
6.02 
6.02 
6.01 
ft  n  i 

0.389964 
.389603 
.389241 

.388880 
.388520 

60 
49 

48 
47 
46 

15 

.578236 

5.15 

.966395 

.86 

.611841 

O.Ul 

.388159 

45 

16 

.578545 

5.14 

.966344 

.86 

.612201 

6.01 
ft  no 

.387799 

44 

17 

.578853 

5.14 

.966292 

.86 

.612561 

o.uu 

.387439 

43 

13 

.679162 

5.14 

.966240 

.86 

.612921 

6.00 

.387079 

42 

19 

.679470 

5.13 
5.13 

.966188 

.86 
.86 

.613281 

6.00 
5.99 

.386719 

41 

20 
21 
22 
23 
24 

9.579777 
.580085 
.580392 
.580699 
.681005 

5.12 
5.12 
6.11 
6.11 

9.966136 
.966085 
.966033 
.965981 
.965929 

.87 
.87 
.87 

.87 

9.613641 
.614000 
.614359 
.614718 
.615077 

5.99 
5.98 
5.98 
5.93 

0.386359 
.386000 
.385641 
.385282 
.384923 

40 
39 
38 
37 
36 

25 
26 
27 
23 
29 

.581312 
.681618 
.681924 
.582229 
.582535 

5.11 
5.10 
5.10 
5.09 
5.09 
5.09 

.965876 
.965824 
.965772 
.965720 
.965668 

.87 
.87 
.87 
.87 
.87 
.87 

.615435 
.615793 
.016151 
.616509 
.616867 

5.97 
5.97 
6.97 
5.96 
5.96 
5.96 

.384565 
.384207 
.383849 
.383491 
.383133 

35 
34 
33 
32 
31 

30 
31 
32 
33 
34 

9.582840 
.633145 
.683449 
.683754 
.584058 

5.08 
5.08 
5.07 
5.07 

9.965615 
.965563 
.965511 
.965458 
.965406 

.87 
.87 
.87 
.87 

U.617224 
.617582 
.617939 
.618295 
.618652 

5.95 
5.95 
5.95 
5.94 

0.382776 
.382418 
.382061 
.381705 
.381348 

30 
29 

28 
27 
26 

35 

.584361 

5.06 

.965353 

.88 

.619008 

5.94 

.380992 

25 

36 

.584665 

5.06 

.965301 

.88 

.619364 

5.94 

.380636 

24 

37 

.584968 

5.06 

.965248 

.88 

.619720 

5.93 

.380280 

23 

33 

.585272 

5.05 

.965195 

.88 

.620076 

5.93 

.379924 

22 

39 

.585574 

5.05 
5.04 

.965143 

.88 
.88 

.620432 

5.93 
5.92 

.379568 

21 

40 

9.585877 

9.965090 

9.620787 

0.379213 

20 

41 

.586179 

5.04 

.965037 

.88 

.621142 

5.92 

.378858 

19 

42 

.586482 

5.04 

.964984 

.88 

.621497 

5.92 

.378503 

18 

43 

.586783 

5.03 

.964931 

.88 

.621852 

5.91 

.378148 

17 

44 

.587085 

5.03 

.964879 

.88 

.622207 

5.91 

.377793 

16 

45 
46 

.587386 

.587688 

5.02 
5.02 

.964826 
.964773 

.88 
.83 

.622561 
.622915 

6.91 
5.90 

.377439 
.377085 

15 

14 

47 

.587939 

5.01 

.964720 

.88 

.623269 

5.90 

.376731 

13 

43 

.588289 

6.01 

.964666 

.88 

623623 

5.90 

376377 

12 

49 

.588590 

5.01 
5.00 

.964613 

.89 
.89 

.623976 

5.89 
5.89 

.376024 

11 

50 

9.588890 

9.964560 

9.624330 

r  on 

0.375670 

10 

51 

.589190 

5.00 

.964507 

.89 

.624683 

o.oy 

.375317 

9 

52 

.589489 

4.99 

.964454 

.89 

.625036 

5.88 

.374964 

8 

53 

.589789 

4.99 

.964400 

.89 

.625388 

5.88 

e  OQ 

.374612 

7 

64 

.590088 

4.99 

.964347 

.89 

.625741 

O.OO 
e  Q» 

.374259 

6 

65 

.590387 

4.98 

.964294 

.89 

.626093 

o.o/ 

.373907 

5 

66 

.590686 

4.98 

.964240 

.89 

.626445 

5.87 

.373555 

4 

67 

58 

.590984 
.591282 

4.97 
4.97 

.964187 
.964133 

.89 
.89 

.626797 
.627149 

6.87 
6.86 

C  Ofl 

.373203 
.372851 

3 
2 

59 

.591580 

Ant* 

.964080 

.89 

.627501 

O.OO 

{•  0<» 

.372499 

I 

60 

.591878 

4.96 

.964026 

.89 

.627852 

u.OO 

.372148 

0 

M. 

Cotdae. 

D.  1". 

Sine. 

D.  1". 

Cotaug. 

D.  1". 

Taug. 

M. 

67° 


COSINES,  TANGENTS,  AND  COTANGENTS.        263 

330                                                 166Q 

ft 

Sine. 

D.  1". 

Cosine. 

D.F. 

Tang. 

D.  F. 

Cotang. 

M. 

0 

2 
3 

4 

6 

7 
8 
9 

9.591878 
.692176 
.592473 
.692770 
.593067 
.693363 
.693659 
.693955 
.594251 
.694547 

4.96 
4.95 
4.95 
4.95 
4.94 
4.94 
4.93 
4.93 
4.93 
4.92 

9.9640*0 
.963972 
.963919 
.963865 
.963811 
.963757 
.963704 
.963650 
.963596 
.963542 

.89 
.89 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 

9.627852 
.628203 
.628554 
.628905 
.629255 
.629606 
.629956 
.630306 
.630656 
.631005 

5.85 

5.85 
5.85 
5.84 
5.84 
6.84 
5.83 
5.83 
5.83 
5.82 

0.372148 
.371797 
.371446 
.371095 
.370745 
.370394 
.370044 
.369694 
.369344 
.368995 

60 
59 
68 
67 
56 
55 
54 
63 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 

9.594842 
.595137 
.595432 
.595727 
.596021 
.596315 
.696609 
.596903 

4.92 
4.91 
4.91 
4.91 
4.90 
4.90 
4.89 

4  ftQ 

9.963488 
.963434 
.963379 
.963325 
.963271 
.963217 
.963163 
.963108 

.90 
.90 
.90 
.90 
.90 
.90 
.91 

Ql 

9.631355 
.631704 
.632053 
.632402 
.632750 
.633099 
.633447 
.633795 

5.82 
6.82 
6.81 
6.81 
5.81 
6.80 
6.80 

c  on 

0.368645 

.368296 
.367947 
.367598 
.367250 
.366901 
.366553 
.366205 

60 
49 

48 
47 
46 
45 
44 
43 

18 
19 

.697196 
.697490 

4.89 
4.83 

.963054 
.962999 

.91 
.91 

.634143 
.634490 

6.79 
6.79 

.365857 
.365510 

42 
41 

20 
21 
22 
23 
24 

9.597783 
.598075 
.698368 
.69866C 
.698952 

4.88 
4.88 
4.87 
4.87 

A  aa 

9.962945 
.962890 
.962836 
.962781 
.962727 

.91 
.91 
.91 
.91 

Ql 

9.634838 
.635185 
.635532 
.635879 
.636226 

6.79 

6.78 
5.78 
6.78 

c  70 

0.365162 
.364815 
.364468 
.364121 
363774 

40 
39 
38 
37 
36 

26 
20 

1  27 

29 

.699244 
.699536 
.599827 
.600118 
.600409 

4.86 
4.86 
4.85 
4.85 
4.84 

.962672 
.962617 
.962562 
.962508 
.962463 

.91 
.91 
.91 

.91 
.92 

.636572 
.636919 
.637265 
.637611 
.637966 

6.77 
6.77 
6.77 
6.76 
6.76 

.363428 
,363081 
.362735 
.362389 
.362044 

36 
34 
33 
32 
31 

30 
31 

9.600700 
.600990 

4.84 

4  ft! 

9.962398 
.962343 

.92 

Q9 

9.638302 
.638647 

6.76 

C  fK 

0.361698 
.361353 

30 
29 

32 
33 

.601280 
.601570 

4.83 

A  CO 

.902288 
.962233 

.92 

.638992 
.639337 

6.76 

.361008 
.360663 

28 
27 

34 
35 

36 
37 
38 
39 

.601860 
.602150 
.602439 
.602728 
.603017 
.603305 

4.83 
4.82 
4.82 
4.81 
4.81 
4.81 

.962178 
.962123 
.962067 
.962012 
.961957 
.961902 

.92 
.92 
.92 
.92 
.92 
.92 

.639682 
.640027 
.640371 
.640716 
.641060 
.641404 

6.74 
6.74 
5.74 
6.73 
5.73 
6.73 

.360318 
.359973 
.359629 
.359284 
.358940 
.358696 

26 
25 
24 
23 
22 
21 

40 
il 
42 
43 

9.603594 
.603882 
.604170 
.604457 

4.80 
4.80 

4.79 

4  7Q 

9.961846 
.961791 
.961735 
.961680 

.92 
.92 
.92 

QO 

9.641747 

.642091 
.642434 
.642777 

6.73 
5.72 
5.72 

K  79 

0.358253 
.357909 
.357566 
.357223 

20 
19 
18 
17 

44 
45 

.604745 
.605032 

4.79 

A  70 

.961624 
.961569 

.93 

.643120 
.643463 

6.71 

c  71 

.356880 
.356537 

16 
15 

46 
47 

.605319 
.605606 

4.78 

.961513 
.961458 

.93 

.643806 
.644148 

6.71 

.356194 
.355852 

14 
13 

48 

19 

.605892 
.606179 

4.7? 
4.77 

.961402 
.961346 

.93 
.93 
.93 

.644490 
.644832 

6.70 
6.70 

.355510 
.355168 

12 
11 

50 

9.606465 

9.961290 

9.646174 

e  /»o 

0.354826 

10 

51 

52 

.606751 

.607036 

4.76 

4  7fi 

.961235 
.961179 

.93 

.645516 
.645857 

5.69 

*  CO 

.354484 
.354143 

9 

8 

53 

.607322 

961123 

.646199 

.353801 

7 

54 
55 
56 
:  57 
58 
69 
60 

.607607 
.607892 
.608177 
.603461 
.608745 
.609029 
.609313 

4.75 
4.74 
4.74 
4.74 
473 
473 

.961067 
.961011 
.960955 
.960399 
.960843 
.960786 
.960730 

.93 
.93 
.93 
.93 
.94 
.94 
.94 

.646540 
646881 
647222 
.647562 
.647903 
.648243 
.643583 

5.68 
5.68 
6.68 
5.67 
5.67 
5.67 

.353460 
.353119 
.352778 
.352438 
.352097 
.351757 
.351417 

6  1 
6 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.F. 

Sine 

D.  1". 

Co  tang 

D.  F. 

Tang. 

M. 

6G« 


264 

840 


TABLE   XV.       LOGARITHMIC    SINES, 


M. 

Sine. 

D.  1". 

Cosine. 

D.  1«. 

Tang. 

D.I'. 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9.609313 

.609597 
.609880 
.610164 
.610447 
.610729 
.611012 
.611294 
.611576 
.611858 

4.73 
4.72 
4.72 
4.72 
4.71 
4.71 
4.71 
4.70 
4.70 
4.69 

9.960730 
.960674 
.960618 
.960561 
.960505 
.960448 
.960392 
.960335 
.960279 
.960222 

.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 

9.648583 
.648923 
.649263 

.649602 
.649942 
.650281 
.650620 
.650959 
.651297 
.651636 

5.67 
5.66 
5.66 
5.66 
5.65 
5.65 
5.65 
5.64 
5.64 
5.64 

0.351417 
.351077 
.350737 
.350398 
.350058 
.349719 
.349380 
319041 
.348703 
.348364 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 

9.612140 
.612421 

4.69 

d  RQ 

9.960165 
.960109 

.95 

9.651974 
.652312 

5.64 

ff  /»q 

0.348026 
.347688 

50 

49 

12 

.612702 

4.O3 
A  CO 

.960052 

.95 

.652650 

O.DO 

.347350 

48 

13 
14 
15 
16 
17 
18 
19 

.612983 
.613264 
.613545 
.613825 
.614105 
.614385 
.614665 

"I.  Do 

4.68 
4.68 
4.67 
4.67 
4.67 
4.66 
4.66 

.959995 
.959938 
.959882 
.959825 
.959768 
.959711 
.959654 

.95 
.95 
.95 
.95 
.95 
.95 
.95 
.95 

.652988 
.653326 
.653663 
.654000 
.654337 
.654674 
.655011 

5.63 
5.63 
5.62 
5.62 
5.62 
5.62 
5.61 
5.61 

.347012 
.346674 
.346337 
.346000 
.345663 
.345326 
.344989 

47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 

9.614944 
.615223 
.615502 
.615781 
.616060 
.616338 
.616616 

4.65 
4.65 
4.65 
4.64 
4.64 
4.64 

9  959596 
.959539 
.959482 
.959425 
.959368 
.959310 
.959253 

.95 
.95 
.95 
.95 
.96 
.96 

9.655348 
.655684 
.656020 
.656356 
.656692 
.657028 
.657364 

6.61 
5.61 
5.60 
6.60 
5.60 
6.59 

0.344652 
.344316 
.343980 
.343644 
.343308 
.342972 
.342636 

40 
39 
38 
37 
36 
35 
34 

27 

28 
29 

.616894 
.617172 

.617450 

4.63 
4.63 
4.63 
4.62 

.959195 

.959138 
.959080 

.96 
96 
.96 
.96 

.657699 
.658034 
.658369 

5.59 
5.59 
5.58 
5.58 

.342301 
.341966 
.341631 

33 
32 
31 

30 
31 
32 
33 

34 
35 
36 
37 
38 
39 

9.617727 

.618004 
.618281 
.618553 
.618834 
.619110 
.619386 
.619662 
.619938 
.620213 

4.62 
4.61 
4.61 
4.61 
4.60 
4.60 
4.60 
4.59 
4.59 
4.59 

9.959023 
.958965 
.958908 
.958850 
.958792 
.958734 
.958677 
.958619 
.958561 
.958503 

.96 
.96 
.96 
.96 
.96 
.96 
.96 
.97 
.97 
.97 

9.658704 
.659039 
.659373 
.659708 
.660042 
.660376 
660710 
661043 
661377 
,661710 

5.58 
5.58 
5.57 
6.67 
5.57 
5.56 
5.56 
5.56 
5.56 
5.55 

0.341296 
.340961 
.340627 
.340292 
.339953 
.339624 
.339290 
.338957 
.338623 
.338290 

30 
29 

28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 

9.620488 
.620763 
.621038 
.621313 
.621587 
.621861 

4.58 
4.53 
4.58 
4.57 
4.57 

9.958445 
.953387 
.958329 
.958271 
.958213 
.958154 

.97 
.97 
.97 
.97 
.97 

9.662043 
.662376 
.662709 
.663042 
.663375 
.663707 

6.55 
6.55 
6.54 
6.54 

5.54 

0.337957 
.337624 
.337291 
.336958 
.336625 
.336293 

20 
19 
18 
17 
16 
16 

46 
47 

48 
49 

.622135 
.622409 
.622682 
.622956 

4.57 
4.56 
4.56 
4.56 

4.55 

.958096 
.958038 
.957979 
.957921 

.97 
.97 
.97 
.97 
.97 

664039 
.664371 
.664703 
.665035 

5.54 
5.53 
5.53 
5.53 
5.53 

335961 
335629 
.335297 
.334965 

14 
13 
12 
11 

50 
51 

9.623229 
.623502 

4.55 

9.957863 
.957804 

.97 

9.665366 
.665698 

5.52 

0.334634 
.334302 

10 
9 

52 
53 

.623774 
.624047 

4.54 
4.54 

.957746 
.957687 

.98 
.98 

.666029 
.666360 

5.52 
5.52 

.333971 
.33-3640 

8 
7 

54 

.624319 

4.54 

.957628 

.98 

.666691 

5.61 

.333309 

6 

55 

.624591 

4.53 

.957570 

.98 

667021 

5.51 

332979 

5 

56 

.624863 

4.53 

A.  £Q 

.957511 

.98 

QQ 

.667352 

5.51 

K  ei 

.332648 

4 

57 

.625135 

1.OO 

.957452 

.yo 

.667682 

D.OJ 

.332318 

3 

68 
59 

.625406 
625677 

4.52 
4.52 

.957393 
.957335 

.98 
.98 

.668013 
.668343 

5.50 
5.50 

.331987 
.331657 

2 

60 

.625948 

4.52 

.957276 

.98 

.668673 

5.50 

.331327 

0 

M. 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  1".   Tang. 

II. 

1140 


COSINES,    TANGENTS,    AND   COTANGENTS. 


265 


M 

Blue. 

D.l". 

Cosine. 

D  1" 

Tung. 

D.1". 

Ootaiig. 

M. 

"T 

i 

3 

9.625948 
.626219 
.626490 
.626760 

4.61 
4.61 
4.51 

A  RA 

9.957276 
.957217 
.957168 
.957099 

.98 
.98 

.98 

QO 

9.668673 
.669002 
.669332 
.669661 

6.50 
6.49 

6.49 

0.331327 
.330998 
.330668 
.330339 

~6JT 
69 
68 
67 

4 

6 

7 

.627030 
.627300 
.627670 
.627840 

4.OU 

4.60 

4.50 
4.49 

.957040 
.956981 
.956921 
.956862 

•vQ 

.99 
.99 
.99 
99 

.669991 
.670320 
.670649 
.670977 

6.49 
5.49 
6.48 
6.48 

K  dfl 

.330009 
.329680 
.329351 
.329023 

66 
65 
54 
63 

8 
9 

.628109 
.628378 

149 
4.48 

.956803 
.956744 

'99 

.671306 
.671635 

O.4O 

6.47 
6.47 

.328694 
.328365 

52 
61 

1C 

11 

9J62S647 
.628916 

4.48 

9.956684 
.956625 

.99 

9.671963 
.672291 

5.47 

0.328037 
.327709 

60 
49 

12 

.629185 

4.48 

.956566 

.99 

QQ 

.672619 

6.47 

R  AK. 

.327381 

48 

13 

.629453 

I'lr 

.956506 

••V 

.672947 

O.3O 

.327053 

47 

14 

.629721 

J*£ 

.956447 

.99 

.673274 

6.46 

.326726 

46 

16 
16 

,629989 
.630257 

4.47 
4.46 

.956387 
.956327 

.99 
99 

.673602 
.673929 

6.46 
6.46 

.326398 
.326071 

45 
44 

17 
18 
19 

.630524 
.630792 
.631059 

4.46 
4.46 
4.45 
4.45 

956268 
.956208 
.966148 

.99 
.99 
1.00 
1.00 

.674257 
.674584 
.674911 

6.45 
5.46 
6.45 
6.46 

.325743 
.325416 
.325089 

43 
42 
41 

20 
21 
22 
23 
24 
26 
26 
27 
28 
29 

9.631326 
.631593 
.631859 
.632125 
.632392 
.632658 
.632923 
.633189 
.633454 
.633719 

4.46 
4.44 
4.44 
4.44 
4.43 
4.43 
4.43 
4.42 
4.42 
4.42 

9.956089 
.956029 
.955969 
.955909 
.955849 
.955789 
.955729 
.955669 
.955609 
.956548 

1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 

9.676237 
.675564 
.675890 
.676217 
.676543 
.676869 
.677194 
.677620 
.677846 
.678171 

6.44 
5.44 
5.44 
5.44 
6.43 
6.43 
6.43 
6.42 
6.42 
6.42 

0.324763 
.324436 
.324110 
.323783 
.323457 
.323131 
.322806 
.322480 
.322154 
321829 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 

9.633984 
.634249 

4.41 

9.955488 
.955428 

1.00 

9.678496 
.678821 

6.42 

0.321504 
.321179 

30 
29 

32 
33 

34 
36 
36 
37 

38 
39 

.634514 
.634778 
.635042 
.635306 
.635570 
.635834 
.636097 
.636360 

4.41 
4.41 
4.40 
4.40 
4.40 
4.39 
4.39 
4.39 
4.38 

.955368 
.955307 
.955247 
.955186 
.955126 
.955065 
.955005 
.954944 

1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 

.679146 
.679471 
.679795 
.680120 
.680444 
.680768 
.681092 
.681416 

6.41 
6.41 
6.41 
5.41 
6.40 
6.40 
6.40 
6.40 
6.39 

.320854 
.320529 
.320205 
.319880 
.319556 
.319232 
.318909 
.318534 

28 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
43 

44 
45 

46 
47 

48 
49 

9.636623 
.636886 
.637148 
.637411 
.637673 
.637935 
.638197 
.638458 
.638720 
.638981 

4.38 
4.38 
4.37 
4.37 
4.37 
4.36 
4.36 
4.36 
4.35 
4.36 

9.954883 
.954823 
.954762 
.954701 
.954640 
.954579 
.954518 
.954457 
.954396 
.954335 

1.01 
1.01 
1.01 
101 
1.02 
1.02 
.02 
1.02 
.02 
.02 

9.681740 
.682063 
.682387 
.682710 
.683033 
.683356 
.683679 
.684001 
.684324 
.684646 

5.39 
5.39 
5.39 
5.38 
6.38 
5.38 
5.38 
6.37 
6.37 
6.37 

0.318260 
.317937 
.317613 
.317290 
.316967 
.316644 
.316321 
315999 
.315676 
.315354 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
61 

9.639242 
.639503 

4.35 

9.954274 
.954213 

.02 

9.684968 
.685290 

6.37 

0.315032 
.314710 

10 
9 

52 
53 
54 
55 
56 
57 
58 
59 
60 

.639764 
.640024 
.640284 
.640544 
.640804 
.641064 
.641324 
.641583 
.641842 

4.34 
4.34 
4.34 
4.33 
4.33 
4.33 
4.32 
4.32 
4.32 

.954152 
.954090 
.954029 
.953968 
.953906 
.953845 
.953783 
.953722 
.953660 

.02 
.02 
.02 
.02 
.02 
.02 
.03 
.03 
1.03 

.685612 
.685934 
.686255 
.686577 
.686898 
.687219 
687540 
687861 

5.36 
5.36 
5.36 
5.36 
5.35 
6.35 
5.35 
5.35 
5.35 

.314388 
.314066 
.313745 
.313423 
.313102 
.312781 
.312460 
.312139 
.311818 

8 
7 
6 
6 
4 
3 
2 
t 
0 

M 

Cosine 

D  1" 

Sine 

D.  1". 

Cotang* 

D.I'. 

;  •   JB 

Tug. 

t  = 

M. 

64° 


266           TABLE  XV.   LOGARITHMIC  SINES, 

460                                                 15* 

H. 

Sine.  1  D.I". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

V.  - 

0 

2 
3 

4 
5 
6 
7 
8 
9 

9.64184* 
.642101 

.642360 
.642618 
.642877 
.643135 
.643393 
.643650 
.643908 
.644165 

4.32 
4.31 
4.31 
4.31 
4.30 
4.30 
4.30 
4.29 
4.29 
4.29 

9.953660 
.953599 
.953537 
.953475 
.953413 
.953352 
.953290 
.953228 
.953166 
.953104 

1.03 
1.03 
1.03 
1.03 
1.03 
1.03 
1.03 
103 
1,03 
1.03 

9.688182 
.688502 
.688823 
.689143 
.689463 
.689783 
.690103 
.690423 
.690742 
.691062 

5.34 
5.34 
5.34 
5.34 
5.33 
5.33 
5.33 
5.33 
6.32 
6.32 

0.311818 
.31  1498 
.311177 
,310857 
.310537 
.310217 
.309897 
.309577 
309258 
.308938 

60 
59 
58 
57 
56 
55 
64 
53 
52 
61 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9.644423 
.644680 
.644936 
.645193 
.645450 
.645706 
.645962 
.646218 
.646474 
.646729 

4.28 
4.28 
4.28 
4.27 
4.27 
4.27 
4.26 
4.26 
4.26 
4.26 

9.953042 
.952980 
.952918 
.952855 
.952793 
.952731 
.952669 
.952606 
.952544 
.952481 

1.03 
1.04 
1.04 
1.04 
1.04 
1.04 
1.04 
1.04 
1.04 
1.04 

9.691381 
.691700 
.692019 
.692338 
.692656 
.692975 
.693293 
.693612 
.693930 
.694248 

5.32 
6.32 
5.31 
5.31 
5.31 
6.31 
5.30 
5.30 
6.30 
6.30 

0.308619 
.308300 
.307981 
,307662 
3Cf7344 
.307025 
.306707 
.306388 
.306070 
.305752 

60 
49 
43 
47 
l« 
45 
44  ' 
43 
42  1 
41 

20 
21 
22 

9.646984 
.647240 
.647494 

4.25 

4.25 

9.952419 
.952356 
.952294 

1.04 
1.04 

1  ClA 

9.694566 
.694883 
.695201 

6.29 
5.29 

0.305434 
.305117 
.304799 

40  ' 
39 
33 

23 
24 
25 
26 
27 
28 
29 

.647749 
.648004 
.648253 
.648512 
.648766 
.649020 
.649274 

4.25 
4.24 
4.24 
4.24 
4.23 
4.23 
4.23 
4.22 

.952231 
.952168 
.952106 
.952043 
.951980 
.951917 
.951854 

1.04 
1.05 
1.05 
1.05 
1.05 
105 
1.05 

.695518 
.695836 
.696153 
.696470 
.696787 
.697103 
.697420 

6.29 
5.29 
6.28 
6.28 
6.2S 
5.28 
6.27 

.304482 
.304164 
.303847 
303530 
.303213 
.302897 
.302580 

37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.649527 
.649781 
.650034 
.650287 
.650539 
.650792 
.651044 
.651297 
.651549 
.651800 

4.22 
4.22 
4.22 
4.21 
4.21 
4.21 
4.20 
4.20 
4.20 
4.19 

9.951791 
.951728 
951665 
.951602 
.951539 
.951476 
.951412 
.951349 
.951286 
.951222 

1.05 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 
1.06 
1.06 
1.06 

9.697736 
.698053 
.698369 
.698685 
.699001 
.699316 
.699632 
.699947 
.700263 
.700578 

6.27 
5.27 
5.27 
5.26 
6.26 
6.26 
6.26 
6.26 
5.25 
5.25 

0.302264 
.301947 
.301631 
.301315 
.300999 
.300684 
.300363 
.300053 
.299737 
.299422 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 

9.652052 
.652304 

4.19 

9.951159 
.951096 

1.06 

1  HA 

9.700893 
.701208 

5.26 

0.299107 
.298792 

20 
19 

42 

43 
44 
45 
46 

.652555 
.652806 
.653057 
.653308 
.653558 

4.19 
4.18 
4.18 
4.18 
4.18 

.951032 
.950968 
.950905 
.950841 
.950778 

1.06 
1.06 
1.06 
1.06 

1  DA 

.701523 
.701837 
.702152 
.702466 
.702781 

5.24 
5.24 
5.24 
5.24 

.298477 
.298163 
.297848 
.297534 
.297219 

18 
17 
16 
16 

14 

47 

.653808 

4.17 

.950714 

.703095 

.296905 

13 

48 
49 

654059 
654309 

4.17 
4.17 
4.16 

.950650 
.950586 

1.06 
1.06 

.7034-^9 
.703722 

5.23 
5.23 

.296591 
.296278 

12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

9.654558 
.654808 
.655058 
.655307 
.655556 
.655805 
.656054 
.656302 
.656551 
.656799 
.657047 

4.16 
4.16 
4.15 
4.15 
4.15 
4.15 
4.14 
4.14 
4.14 
4.13 

9.950522 
.950458 
.950394 
.950330 
.950266 
.950202 
.950138 
.950074 
.950010 
.949945 
.949881 

1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 

9.704036 
.704350 
.704663 
.704976 
.705290 
.705603 
.705916 
.706228 
.706541 
.706854 
,707166 

5.23 
5.22 
5.22 
5.22 
6.22 
6.22 
6.21 
521 
5.21 
6.21 

0.295964 
.295650 
.295337 
.295924 
.294710 
.294397 
.294084 
.293772 
.293459 
.293146 
.292834 

10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 

M. 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Ootang. 

D.  1". 

Tang 

M. 

1160 


COSINES,  TANGENTS,  AND  COTANGENTS.        26? 

«y°                           i5» 

M. 

Slue. 

D.  1". 

Cosine. 

D.I" 

Tang. 

D.  1". 

Cotang. 

M. 

0 

2 
3 
4 

5 
6 
7 

c 

9.657047 
.657295 
.657542 
.657790 
.658037 
.658284 
.658531 
.658778 
659025 

4.13 
4.13 
4.12 
4.12 
4.12 
4.12 
4.11 
4.11 

9.949881 
.949816 
.949752 
.949688 
.949623 
.949558 
.949494 
.949429 
.949364 

1.07 

1.07 
1.07 
1.08 
1.03 
1.08 
1.08 
1.08 

9.707166 
.707478 
707790 
.708102 
.708414 
.708726 
.709037 
.709349 
.709660 

5.20 
5.20 
5.20 
5.20 
5.20 
5.19 
5.19 
5.19 

0.292834 
.292522 
.292210 
.291898 
.291586 
.291274 
.290963 
.290651 
.290340 

60 
69 
58 
57 
56 
65 
54 
53 
52 

9 

i  .659271 

4.10 

.949300 

1.08 

709971 

5.19 
5.18 

.290029 

51 

10 

9.659517 

4  10 

9.949235 

1QQ 

9.710282 

s  ift 

0.289718 

50 

a 

12 
13 
14 
15 
16 
17 
18 
19 

.659763 
.660009 
.660255 
.660501 
.660746 
.660991 
.661236 
.661481 
.661726 

4.10 
4.10 
4.09 
4.09 
4.09 
4.08 
4.08 
4.08 
4.08 

.949170 
.949105 
.949040 
.948975 
.948910 
.948845 
.948780 
.948715 
.948650 

1.08 
1.08 
1.08 
1.08 
1.08 
1.09 
1.09 
1.09 
1.09 

.710593 
.710904 
.711215 
711525 
.711836 
.712146 
.712456 
.712766 
.713076 

6.18 
6.18 
6.18 
5.17 
5.17 
6.17 
6.17 
5.17 
5.16 

.289407 
.289096 
.288785 
.288476 
.288164 
.287854 
.287544 
.287234 
.286924 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

9.661970 
.662214 
.662459 
.662703 
.662946 
.663190 
.663433 
.663677 
.663920 
.664163 

4.07 
4.07 
4.07 
4.06 
4.06 
4.06 
4.05 
4.05 
4.05 
4.05 

9.948584 
.948519 
.948454 
.948388 
.948323 
.948257 
.948192 
.948126 
.943060 
.947996 

1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.10 

9.713386 
.713696 
.714005 
.714314 
.714624 
.714933 
.715242 
.715551 
.715860 
.716168 

5.16 
6.16 
5.16 
6.15 
5.15 
5.15 
5.15 
6.15 
6.14 
6.14 

0.286614 
.286304 
.285995 
.285686 
.285376 
.285067 
.284758 
.284449 
.284140 
.283832 

40 
39 
38 
37 
36 
36 
34 
33 
33 
31 

30 
81 
32 
33 
84 
35 
36 
37 
38 
39 

9.664406 
.664648 
.664891 
.635133 
.665375 
.665617 
.665859 
.666100 
.666342 
.666533 

4.04 
4.04 
4.04 
4.03 
4.03 
4.03 
4.03 
4.02 
4.02 
4.02 

9:947929 
.947863 
.947797 
.947731 
.947665 
.947600 
.947533 
.947467 
.947401 
.947335 

1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 

9.716477 

.716785 
.717093 
.717401 
.717709 
.718017 
.718325 
.718633 
.718940 
.719248 

5.14 
6.14 
5.14 
5.13 
5.13 
5.13 
5.13 
5.13 
6.12 
5.12 

0.283523 

.283215 
.282907 
.282599 
.282291 
.281983 
.281675 
.281367 
.281060 
.280752 

30 
29 
28 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
1  43 
44 
45 
46 
47 
43 
49 

9.666824 
.667065 
.667305 
.667546 
.667786 
.668027 
.668267 
.663506 
.668746 
.668936 

4.01 

4.01 
4.01 
4.01 
4.00 
4.00 
4.00 
3.99 
3.99 
3.99 

9.947269 
.947203 
.947136 
.947070 
.947004 
.946937 
.946871 
.946804 
.946738 
.946671 

1.10 

1.11 
1.11 
1.11 
1.11 
1.11 
1.11 
1.11 
1.11 
1.11 

9.719555 
.719862 
.720169 
720476 
.720783 
.721089 
.721396 
.721702 
.722009 
.722315 

5.12 
5.12 
5.11 
5.11 
6.11 
5.11 
5.11 
6.10 
5.10 
5.10 

0.280445 
.280138 
.279831 
.279524 
.279217 
.278911 
.278604 
.278298 
.277991 
.277685 

20  ! 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
66 
57 
68 
69 
60 

9.669225 
.669464 
669703 
.669942 
.670181 
.670419 
.670653 
.670396 
.671134 
671372 
.671609 

3.99 
3.93 
3.98 
3.93 
3.98 
3.97 
3.97 
3.97 
3.96 
3.96 

9.946604 
.946538 
.946471 
.946404 
.946337 
.946270 
,946203 
.946136 
.946069 
.946002 
.945935 

1.11 
1.11 
1.11 
1.11 

1.12 
1.12 
1.12 
1.12 
1.12 
1  12 

9.722621 
.722927 
.723232 
.723538 
.723844 
.724149 
.724454 
.724760 
.725065 
.725370 
.725674 

5.10 
6.10 
6.09 
5.09 
5.09 
5.09 
5.09 
5.08 
5.08 
5.08 

0.277379 
.277073 
.276768 
.276462 
.276156 
.275851 
.275546 
.275240 
.274935 
.274630 
.274326 

10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 

I  M 

Cosine. 

D.  1". 

Sine. 

D  1". 

Cotang. 

D.I'. 

T*ng. 

M. 

69Q 


268           TABLE  XV.   LOGARITHMIC  SINES, 

880                                             1BJ 

M. 

Bice. 

D.  1". 

Cosine. 

D.  1". 

Tang.  • 

D.  1". 

Cotacg. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9.671609 
.671847 
.672084 
.672321 
.672558 
.672795 
.673032 
.673268 
.673505 
.673741 

3.96 
3.96 
3.95 
3.95 
3.95 
3.94 
3.94 
3.94 
3.94 
3  93 

9.945935 
.945868 
.945800 
.945733 
.945666 
.945598 
.945531 
.945464 
.945396 
.945328 

1.12 
1.12 
1.12 
1.12 
1.12 
1.12 
1.12 
1.13 
1.13 
1.13 

9.725674 
.725979 
.726284 

.726588 
.726892 
.727197 
.727501 
.727805 
.728109 
.728412 

6.08 
5.08 
5.07 
5.07 
6.07 
6.07 
5.07 
5.06 
6.06 
5.06 

0.274326 
.274021 
.273716 
.273412 
.273108 
.272803 
.272499 
.272195 
.271891 
.271588 

60 
59 
58 
57 
56 
5f 
54 
53 
52 
51 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9.673977 
.674213 
.674448 
.674684 
.674919 
.675155 
.675390 
.675624 
.675859 
.676094 

3.93 
3.93 
3.93 
3.92 
3.92 
3.92 
3.91 
3.91 
3.91 
3  91 

9.945261 
.945193 
.945125 
.945058 
.944990 
944922 
.944854 
944786 
.944718 
.944650 

1.13 
1.13 
1.13 
1.13 
,.13 
1.13 
1.13 
1.13 
1.13 
1.13 

9.728716 
.729020 
729323 
729626 
729929 
730233 
730535 
730838 
731141 
731444 

5.06 
5.06 
5.05 
5.05 
5.06 
5.05 
5.06 
5.05 
5.04 
5.04 

0.271284 
.270980 
.270677 
.270374 
.270071 
.269767 
.269465 
.269162 
.268859 
.268556 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
26 
26 
27 
28 
29 

9.676328 
.676562 
.676796 
.677030 
.677264 
.677498 
.677731 
.677964 
.678197 
.678430 

3.90 
3.90 
3.90 
3.90 
3.89 
3.89 
3.89 
3.88 
3.88 
388 

9.944582 
.944514 
.944446 
.944377 
.944309 
.944241 
.944172 
.944104 
.944036 
.943967 

1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 

9.731746 
.732048 
.732351 
.732653 
.732955 
.733257 
.733558 
.733860 
.734162 
.734463 

5.04 
5.04 
6.04 
5.03 
6.03 
6.03 
6.03 
5.03 
5.02 
5.02 

0.268254 
.267952 
.267649 
.267347 
.267045 
.266743 
.266442 
.266140 
.265838 
.265537 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.678663 

.678895 
.679128 
.679360 
.679592 
.679824 
.680056 
.680288 
.680519 
.680750 

3.88 
3.87 
3.87 
3.87 
3.87 
3.86 
386 
3.86 
3.86 
385 

9.943899 
.943830 
.943761 
.943693 
.943624 
.943555 
.943486 
.943417 
.943348 
.943279 

1.14 
1.14 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1  16 

9.734764 
.735066 
.735367 
.735668 
.735969 
.736269 
.736570 
.736870 
.737171 
.737471 

6.02 
6.02 
5.02 
5.01 
6.01 
6.01 
5.01 
6.01 
6.01 
6.00 

0.265236 
.264934 
.264633 
.264332 
.264031 
.263731 
.263430 
.263130 
.262829 
.262529 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
18 
49 

9.680982 
.681213 
.681443 
.681674 
.681905 
682135 
.682365 
.682595 
.682825 
.683055 

3.85 
3.85 
3.84 
3.84 
3.84 
3.84 
3.83 
3.83 
3.83 
383 

9.943210 
.943141 
.943072 
.943003 
.942934 
.942864 
.942795 
.942726 
.942656 
.942587 

1.15 
1.16 
1.15 
1.15 
1.15 
1.16 
1.16 
1.16 
1.16 
1  16 

9.737771 
.738071 
.738371 
.738671 
.738971 
.739271 
.739570 
.739870 
.740169 
.740468 

5.00 
5.00 
5.00 
6.00 
4.99 
4.99 
4.99 
4.99 
4.99 
4.98 

0.262229 
.261929 
.261629 
.261329 
.261029 
.260729 
.260430 
.260130 
.259831 
.25953, 

20 
19 
18 
17 
J6 
15 
14 
13 
12 
11 

60 
61 
52 
53 
64 
65 
66 
67 
63 
59 
60 

9.683284 
.683514 
.633743 
.683972 
.684201 
.684430 
.684658 
.684887 
.685115 
.685343 
.685571 

3.82 

3.82 
3.82 
3.82 
3.81 
3.81 
3.81 
3.80 
3.80 
3.80 

9.942517 
.942448 
.942378 
.942308 
.942239 
.942169 
.942099 
.942029 
.941959 
.941889 
.941819 

1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.17 
1.17 
1.17 

9.740767 
.741066 
.741365 
.741664 
.741962 
.742261 
.742559 
.742858 
.743156 
.743454 
.743752 

4.98 
4.98 
4.98 
4.98 
4.98 
4.97 
4.97 
4.97 
4.97 
4.97 

0.259233 
.258934 
.258635 
.258336 
.258038 
.257739 
.257441 
.257142 
.256844 
.256546 
.256248 

a 

8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Goedue. 

D  1". 

Slue. 

D.  1". 

Cotang. 

D.I". 

Tang. 

— 

M. 

118° 


COSINES,    TANGENTS,    AND   COTANGENTS. 


269 


M 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Gotang. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9.685571 
685799 
.686027 
686254 
.686482 
.686709 
.686936 
687163 
.687389 
.687616 

3.80 
3.79 
3.79 
3.79 
3.79 
3.78 
3.78 
3.78 
3.78 
3.77 

9.941819 
.941749 
.941679 
.941609 
.941539 
.941469 
.941398 
.941328 
.941258 
.941187 

1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 

9.743752 
.744050 
.744348 
.744645 
.744943 
.745240 
.745538 
.745835 
.746132 
.746429 

4.96 
4.96 
4.96 
4.96 
4.96 
4.96 
4.95 
4.95 
4.95 
4.95 

0.256248 
.255950 
.255652 
.255355 
.255057 
.254760 
.254462 
.254165 
.253863 
.253571 

60 
69 
68 
57 
68 
55 
64  | 
53 
62 
61 

1C 
11 

12 
13 
14 
15 
16 
17 
18 
19 

9.687843 
.688069 
.688295 
.688521 
.688747 
.688972 
.689198 
.689423 
.689648 
.689873 

3.77 
3.77 
3.77 
3.76 
3.76 
3.76 
3.76 
3.75 
3.75 
3.75 

9.941117 
.941046 
.940975 
.940905 
.940834 
.940763 
.940693 
.940622 
.940551 
.940480 

1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 

9.746726 
.747023 
.747319 
.747616 
.747913 
.748209 
.748505 
.748801 
.749097 
.749393 

4.95 
4.95 
4.94 
4.94 
4.94 
4.94 
4.94 
4.93 
4.93 
4.93 

0.253274 
.252977 
.252681 
.252384 
.252087 
.251791 
.251495 
.251199 
.250903 
.250607 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.690098 
.690323 
.690548 
.690772 
.690996 
.691220 
,691444 
.691668 
.691892 
.692115 

3.75 
3.74 
3.74 
3.74 
3.74 
3.73 
3.73 
3.73 
3.73 
372 

9.940409 
.940338 
.940267 
.940196 
.940125 
.940054 
.939982 
.939911 
.939840 
.939768 

1.18 
1.18 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 

9.749689 
.749985 
.750281 
.750576 
.750872 
.751167 
.751462 
.751757 
.752052 
.752347 

4.93 
4.93 
4.93 
4.92 
4.92 
4.92 
4.92 
4.92 
4.92 
4.91 

0.250311 
.250015 
.249719 
.249424 
.249128 
.248833 
.248538 
.248243 
.247948 
.247653 

40 
39 
38 
37 
3d 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.692339 

.692562 
.692785 
.693008 
.693231 
.693453 
.693676 
.693898 
.694120 
.694342 

3.72 
3.72 
3.72 
3.71 
3.71 
3.71 
3.71 
.  3.70 
3.70 
3.70 

S.939697 
.939625 
.939554 
.939482 
.939410 
.939339 
.939267 
.939195 
.939123 
.939052 

1.19 
1.19 
1.19 
1.19 
1.19 
1.20 
1.20 
1.20 
1.20 
1.20 

9.752642 
.752937 
.753231 
.753526 
.753820 
.754115 
.754409 
.754703 
.754997 
.755291 

4.91 
4.91 
4.91 
4.91 
4.91 
4.90 
4.90 
4.90 
4.90 
4.90 

0.247368 

.247063 
.246769 
.246474 
.246180 
.245885 
.245591 
.245297 
.245003 
.244709 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 

9.694564 
.694786 

3.70 

9.933980 

.938908 

1.20 

9.755585 
.755878 

4.89 

0.244415 
.244122 

20 
19 

42 
43 
44 
45 
46 
47 
48 
49 

.695007 
.695229 
.695450 
.695671 
.695892 
.696113 
.696334 
.696554 

3.69 
3.69 
3.69 
3.69 
3.68 
3.68 
3.68 
3.68 
3.67 

.938836 
.938763 
.938691 
.933619 
.933547 
.938475 
,938402 
.938330 

1.20 
1.20 
1.20 
1.20 
1.20 
1.20 
1.21 
1.21 
1.21 

.756172 
.756465 
.756759 
.757052 
.757345 
.757638 
.757931 
.758224 

4.89 
4.89 
4.89 
4.89 
4.89 
4.88 
4.88 
4.88 
4.88 

.243828 
.243535 
.243241 
.242948 
.242655 
.242362 
.242069 
.241776 

18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.696775 
.696995 
.697215 
.697435 
.607654 
.697874 
.698094 
.698313 
.698532 
.698751 
.698970 

3.67 
3.67 
3.67 
3.66 
3.66 
3.66 
3.66 
3.65 
3.65 
3.65 

9.938258 
.938185 
.938113 
,938040 
.937967 
.937895 
.937822 
.937749 
.937676 
.937604 
.937531 

1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.22 

9.758517 
.758810 
.759102 
.759395 
.759687 
.759979 
.760272 
.760564 
.760856 
.761148 
.761439 

4.83 
4.88 
4.87 
4.87 
4.87 
4.87 
4.87 
4.87 
4.86 
4.86 

0.241483 
.241190 

.240893 
.240605 
.240313 
.240021 
.239728 
.239436 
.239144 
.238852 
.238561 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Coelne. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  l'». 

Tnng. 

M. 

600 


270           TABLE  XV.   LOGARITHMIC  SINES, 

800                                             149' 

M. 

Slae. 

D.I*. 

Cofdne. 

D.  1". 

Ttag. 

D.l«. 

Cottmg. 

M 

0 

7 
8 
9 

9.698970 
.699189 
.099407 
.699626 
.699844 
.700062 
.700280 
.700498 
.700716 
.700933 

3.65 
3.64 
3.64 
3.64 
3.64 
3.63 
3.63 
3.63 
3.63 
3.62 

9.937531 
.937458 
.937385 
.937312 
.937238 
.937166 
.937092 
.937019 
.936946 
.936872 

1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 

9.761439 
.761731 

.762023 
.762314 
.762606 
.762897 
.763188 
.763479 
.763770 
.764061 

4.86 
4.86 
4.86 
4.86 
4.86 
4.85 
4.85 
4.85 
4.85 
4.86 

0.238561 
.238269 
.237977 
.237686 
.237394 
.237103 
.236312 
.236521 
.236230 
.235939 

60 
69 
68 
57 
66 
55 
54 
63 
62 
61 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.7C1151 

.701368 
.701585 
.701802 
.702019 
.702236 
.702452 
.702669 
.702885 
.703101 

3.62 
3.62 
3.62 
3.61 
3.61 
3.61 
3.61 
3.60 
3.60 
3.60 

9.936799 
.936725 
.936652 
.936578 
.936505 
.936431 
.936357 
.936284 
.936210 
.936136 

1.22 
1.23 
1.23 
1.23 
1.23 
1.23 
1.23 
1.23 
1.23 
1.23 

9.764352 
.764643 
.764933 
.765224 
.765514 
.765805 
.766095 
.766385 
.766675 
766965 

4.85 
4.84 
4.84 
4.84 
4.84 
4.84 
4.84 
4.83 
4.83 
4.83 

0.235648 
.235357 
.235067 
.234776 
.234486 
.234195 
.233905 
.233615 
.233325 
.233035 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.703317 
.703533 
.703749 
.703964 
.704179 
.704395 
.704610 
.704825 
.705040 
.705254 

3.60 
3.59 
3.59 
3.59 
3.59 
3.59 
3.58 
3.58 
3.58 
3.58 

9.936062 
.935988 
.935914 
.935840 
.935766 
.935692 
.935618 
.935543 
.935469 
.935395 

1.23 
1.23 
1.23 
1.23 
1.24 
1.24 
1.24 
1.24 
1.24 
1.24 

9.767255 
.767646 
.767834 
.768124 
.768414 
.768703 
.768992 
.769281 
.769571 
.769860 

4.83 
4.83 
4.83 
4.82 
4.82 
4.82 
4.82 
4.82 
4.82 
4.82 

0.232745 
.232455 
.232166 
.231876 
.231586 
.231297 
.231008 
.230719 
.230429 
.230140 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.705469 
.705683 
.705898 
.706112 
.706326 
.706539 
.706753 
.706967 
.707180 
.707393 

3.67 
3.67 
3.67 
3.57 
3.56 
3.56 
3.56 
3.56 
3.55 
3.55 

9.935320 
.935246 
.935171 
.935097 
.935022 
.934948 
.934873 
.934798 
.934723 
.934649 

1.24 
1.24 
1.24 
1.24 
1.24 
1.24 
1.25 
1.25 
1.25 
1.26 

9.770148 
.770437 
.770726 
.771016 
.771303 
.771692 
.771880 
.772168 
.772457 
.772745 

4.81 
4.81 
4.81 
4.81 
4.81 
4.81 
4.80 
4.80 
4.80 
4.80 

0.229852 
.229563 
.229274 
.228985 

.228697 
.228408 
.228120 
.227832 
.227543 
.227255 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9707606 

.707819 
.708032 
.708245 
.708458 
.708670 
.708882 
.709094 
.709306 
.709518 

3.55 
3.55 
3.54 
3.54 
3.54 
3.54 
3.54 
3.53 
3.53 
3.53 

9.934574 
.934499 
.934424 
.934349 
.934274 
.934199 
.934123 
.934048 
.933973 
.933898 

1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
1.26 
1.26 

9.773033 
.773321 
.773608 
.773896 
.774184 
.774471 
.774759 
.775046 
.775333 
.775621 

4.80 
4.80 
4.80 
4.79 
4.79 
4.79 
4.79 
4.79 
4.79 
4.78 

0.226967 
.226679 
.226392 
.226104 
.225816 
.225529 
.225241 
.224954 
.224667 
.224379 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 

9.709730 
.709941 
.710153 

3.53 
3.52 

9.933822 
.933747 
.933671 

1.26 
1.26 

1  OA 

9.775908 
.776195 
.776482 

4.78 
4.78 

A  7Q 

0224092 
.223805 
.223518 

10 
9 
8 

53 
54 
55 
56 
57 
58 
59 
60 

.710364 
.710575 
.710786 
.710997 
.711208 
.711419 
.711629 
.711839 

3.52 
3.62 
3.51 
3.51 
3.51 
3.61 
3.51 

.933596 
933520 
933445 
933369 
933293 
933217 
.933141 
.933066 

1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 

.776768 
.777055 
.777342 

.777628 
.777915 
.778201 

.778488 
.778774 

4.78 
4.78 
4.78 
4.77 
4.77 
4.77 
4.77 

.223232 
.222945 
.222658 
.222372 
.222085 
.221799 
.221512 
.221226 

7 
6 
6 
4 
3 
2 
1 
0 

M. 

,:  rr 

Ooelne.  j 

D.1*. 

Sh*. 

D.l» 

Cotang. 

D.l" 

Ttof. 

M. 

LJKP 


COSINES,  TANGENTS,  AND  COTANGENTS.        271 
10                                               1480 

M. 

Bine. 

D.  1» 

Coeine. 

D.  1". 

f*ng. 

D.  1". 

Gotang 

M. 

0 
1 

3 

4 
6 
6 
7 
8 
9 

9.711839 
.712060 
.712260 
.712469 
.712679 
.712889 
.713098 
.713308 
.713517 
.713726 

3.50 
3.60 
3.60 
3.60 
3.49 
3.49 
3.49 
3.49 
3.48 
3.48 

9.933066 
932990 
.932914 
932838 
.932762 
932685 
932609 
.932533 
.932457 
932380 

1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 

9.778774 
.779060 
.779346 
.779632 
.779918 
.780203 
.780489 
.780775 
.781060 
.781346 

4.77 
4.77 
4.77 
4.76 
4.76 
4.76 
4.76 
4.76 
4.76 
4  76 

0.221226 
.220940 
.220654 
.220368 
.220082 
.219797 
.219511 
.219225 
.218940 
.218654 

60 
69 
68 
57 
56 
56 
64 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.713935 
.714144 
.714352 
.714561 
.714769 
.714978 
.715186 
.715394 
.715602 
.716809 

3.48 
3.43 
3.48 
3.47 
3.47 
3.47 
3.47 
3.46 
3.46 
3.46 

9.932304 
.932228 
.932151 
.932075 
.931993 
.931921 
.931845 
.931768 
.931691 
931614 

1.27 
1.27 
1.28 
1.28 
1.28 
1.28 
1.28 
1.28 
1.28 
1.28 

9.781631 
.781916 
.782201 
.782486 
.782771 
.783056 
.783341 
.783626 
.783910 
.784195 

4.76 
4.76 
4.76 
4.75 
4.75 
4.75 
4.75 
4.74 
4.74 
4  74 

0.218369 
218034 
.217799 
.217514 
.217229 
.216944 
.216659 
.216374 
.216090 
.216806 

60 
49 
48 
47 
40 
46 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.716017 
.716224 
.716432 
.716639 
716846 
.717053 
.717259 
.717466 
.717673 
.717879 

3.46 
3.46 
3.45 
3.45 
3.45 
3.45 
3.44 
3.44 
3.44 
3.44 

9.931537 
.931460 
931383 
.931306 
.931229 
.931  152 
931075 
.930998 
.930921 
.930843 

1.23 
1.28 
1.23 
1.28 
1.29 
1.29 
1.29 
1.29 
1.29 
1.29 

9784479 
784764 
785048 
.785332 
.785616 
785900 
.786184 
.786463 
.786762 
.787036 

4.74 
4.74 
4.74 
4.74 
4.73 
4.73 
4.73 
4.73 
4.73 
4  73 

0.215521 
.215236 
.214952 
.214668 
.214384 
.214100 
213816 
.213532 
.213248 
.212964 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
36 
36 
37 
38 
39 

9.718086 
.718291 
.718497 
.718703 
.718909 
.719114 
.719320 
.719525 
.719730 
.719935 

3.43 
3.43 
3.43 
3.43 
3.43 
3.42 
3.42 
3.42 
342 
3.41 

9.930766 
.930688 
930611 
.930533 
.930456 
.930378 
.930300 
.930223 
.930145 
.930067 

1.29 
1.29 
1.29 
1.29 
1.29 
1.29 
1.30 
1.30 
1.30 
1  30 

9.787319 
.787603 

.787880 
.788170 
.788453 
.788736 
.789019 
.789302 
.789585 
.789868 

4.73 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.71 
4  71 

0.212681 
.212397 
.212114 
.211830 
.211647 
211264 
.210981 
.210698 
.210416 
,210132 

30 
29 
28 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
43 
44 
46 
46 
47 
48 
49 

9.720140 
.720345 
.720549 
.720754 
.720958 
.721162 
.721366 
.721570 
.721774 
.721978 

3.41 
3.41 
3.41 
3.41 
3.40 
3.40 
3.40 
3.40 
3.39 
3.39 

9.929989 
.929911 
.929833 
.929755 
.929677 
.929599 
929521 
.929442 
.929364 
.929286 

1.30 
1.30 
1.30 
1.30 
1.30 
1.30 
1.30 
1.31 
1.31 
1  31 

9.790151 
.790434 
.790716 
.790999 
.791281 
.791563 
.791846 
.792128 
.792410 
.792692 

4.71 
4.71 
4.71 
4.71 
4.71 
4.70 
4.70 
4.70 
4.70 

4  ffl 

0.209849 
.209566 
.209284 
.209001 
.208719 
.208437 
.208154 
.207872 
.207590 
.207308 

20 
19 
18 
17 
16 
16 
14 
13 
12 
11 

6C 
61 
62 
63 
64 
65 
66 
57 
68 
69 
60 

9.722181 
.722385 

.722588 
.722791 
.722994 
.723197 
.723400 
723603 
.723805 
.724007 
.724210 

3.39 
3.39 
3.39 
3.38 
3.38 
3.38 
3.38 
3.37 
3.37 
3.27 

9.929207 
.929129 
.929050 
.928972 
.928893 
.928815 
.928736 
928657 
.928578 
.928499 
.928420 

1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.32 

9.792974 
.793256 
.793538 
.793819 
.794101 
794383 
794664 
794946 
795227 
.795508 
.795789 

4.70 
4.70 
4.70 
4.69 
4.69 
4.69 
4.69 
4.69 
4.69 
469 

0.207026 
.206744 
.206462 
.206181 
.205899 
.205617 
.205336 
.205054 
.204773 
.204492 
.204211 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0 

M.  1 

Oorfue.  | 

D.  i". 

Oil* 

D  1". 

Gotaug. 

D.iw. 

Tang 

M 

080 


272           TABLE  XV.   LOGARITHMIC  SINES, 

ft»o                                           14T 

M. 

Sine. 

D.  I". 

Cosine. 

D.  1". 

Tang 

D.  1". 

Cotang. 

M. 

0 
I 

8 
4 

6 
6 

7 
8 
9 

9.724210 
.724412 
.724614 
.724816 
.725017 
.725219 
.726420 
.725622 
.725823 
.726024 

3.37 
3.37 
3.36 
3.36 
3.36 
3.36 
3.36 
3.35 
3.35 
3.35 

9.928420 

.928342 
.928263 
.928183 
.928.04 
.928025 
.927946 
.927867 
.927787 
.927708 

1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 

9.795789 
.796070 
.796351 
.796632 
.796913 
.797194 
.797474 
.797755 
.798036 
.798316 

4.68 
468 
4.68 
4.68 
4.68 
4.68 
4.68 
4.68 
4.67 
4.67 

0.2042H 
.203930 
.203649 
.203368 
.203087 
.202806 
.202526 
.202245 
.201964 
.201684 

60 
59 
58 
57 
56 
55 
54 
53 
52 
61 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9.726225 
.726426 
.726620 
.726827 
.727027 
.727228 
.727428 
.727628 
.727828 
.728027 

3.35 
3.34 
3.34 
3.34 
3.34 
3.34 
3.33 
3.33 
3.33 
3.33 

9.927629 
.927549 
.927470 
.927390 
.927310 
.927231 
.927151 
.927071 
.926991 
.926911 

1.32 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 

9.798596 

.798877 
.799157 
.799437 
.799717 
.799997 
.800277 
.800557 
.800836 
.801116 

4.67 
4.67 
4.67 
4.67 
4.67 
4.66 
4.66 
4.66 
4.66 
4.66 

0.201404 
.201123 
.200843 
.200563 
.200283 
.200003 
.199723 
.199443 
.199164 
.198884 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

9.728227 

q  qq 

9.926831 

1  !W 

9.801396 

4  RR 

0.198604 

40 

21 
22 

.728427 
.728626 

3.32 

q  qg 

.926751 
.926671 

1.33 

Iqq 

801675 
.801955 

4.66 

A  RR 

.198325 
.198045 

39 
38 

23 
24 

26 
26 
27 
28 
29 

728825 
.729024 
.729223 
.729422 
.729621 
.729820 
.730018 

3.32 
3.32 
3.31 
3.31 
3.31 
3.31 
3.31 

.926591 
.926511 
.926431 
.926351 
.926270 
.926190 
.926110 

1.34 
1.34 
1.34 
1.34 
1.34 
1.34 
1.34 

.802234 
802513 
.802792 
.803072 
.803351 
.803630 
.803909 

4.65 
4.65 
4.65 
4.65 
4.65 
4.65 
466 

.197766 
.197487 
.197208 
.196928 
.196649 
.196370 
.196091 

37 
36 
36 
34 
33 
32 
31 

80 
31 
82 
83 
84 
36 
86 
37 
88 
89 

9.730217 
.730416 
.730613 
.730811 
.731009 
.731206 
.731404 
.731602 
.731799 
.731998 

3.30 
3.30 
3.30 
3.30 
3.30 
3.29 
3.29 
3.29 
3.29 
3.28 

9.926029 
.925949 
925868 
.925788 
.925707 
.925626 
.925545 
.925465 
.925384 
.925303 

1.34 
1.34 
1.34 
1.34 
1.35 
1.35 
1.35 
1.35 
1.35 
1.35 

9.804187 
804466 
.804745 
805023 
805302 
.805580 
.805859 
.806137 
.806415 
.a)6693 

4.65 
4.64 
4.64 
4.64 
4.64 
1.64 
4.64 
4.64 
4.64 
4.63 

0.195813 
.195534 
.195255 
.194977 
.194698 
.194420 
.194141 
.193863 
.193586 
.193307 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
46 
46 
47 
48 
49 

9.732193 

.732390 
.732587 
.732784 
.732980 
.733177 
.733373 
.733569 
.733765 
.733961 

3.28 
3.28 
3.28 
3.28 
3.27 
3.27 
3.27 
3.27 
3.27 
3.26 

9.925222 
.925141 
.925060 
.924979 
.924897 
.924816 
.924735 
.924654 
.924572 
.924491 

1.36 
1.35 
1.35 
1.35 
1.35 
1.35 
1.36 
1.36 
1.36 
1.36 

y  806971 
.807249 
,807527 
807805 
.808083 
.808361 
.808638 
.808916 
.809193 
.809471 

4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.G3 
4.G2 
4.62 
4.62 

0.193029 
.192761 
.192473 
.192195 
.191917 
.191639 
.191363 
.19IOW4 
.190807 
.190529 

20 
19 
18 
17 
16 
16 
14 
13 
12 
11 

60 
61 

9.734157 
.734353 

3.26 

q  OR 

9.924409 
.924328 

1.36 

IqR 

9.809748 
.810025 

4.62 

A  RO 

0.190252 
.189975 

10 
9 

62 
63 
64 

66 
66 
67 

.734549 
.734744 
.734939 
.735135 
.735330 
.735525 

3.26 
3.26 
3.25 
3.25 
3.25 

.924246 
.924164 
.924083 
.924001 
.923919 
.923837 

1.36 
1.36 
1.36 
1.36 
1.36 

.810302 
.810580 
.810857 
811134 
.811410 
.811687 

4.C2 
4.62 
4.62 
4.61 
4.61 

.189698 
.189420 
.189143 
.188866 
.188590 
.188313 

8 
7 
6 
6 
4 
8 

68 
69 
60 

.735719 
.735914 
.736109 

3.25 
3.25 
3.24 

923755 
.923673 
.923591 

1.37 
1.37 
1.37 

.811964 
.812241 
,812517 

4.61 
4.61 

.188036 
187769 
.187483 

2 
1 
0 

M. 

Cosine 

D.  1". 

*» 

D.I" 

Ootaug 

D.I" 

Ttag 

M. 

COSINES,    TANGENTS,    AND   COTANGENTS. 


33° 


273 

14O° 


M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  I". 

Cotang. 

M. 

0 
1 
2 
3- 
4 
5 
6 
7 
8 
9 

9.736109 
.736303 
.736498 
.736692 
.736886 
.737080 
.737274 
.737467 
.737661 
.737855 

3.24 
3.24 
3.24 
3.23 
3.23 
3.23 
3.23 
3.23 
3.22 
3.22 

9.923591 
.923509 
.923427 
.923345 
.923263 
.923181 
.923098 
.923016 
.922933 
.922851 

1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.38 

9.812517 
.812794 
.813070 
.813347 
.813623 
.813899 
.814176 
.814452 
.814728 
.815004 

4.61 
4.61 
4.61 
4.61 
4.60 
4.60 
4.60 
4.60 
4.60 
4.60 

0.187483 
.187206 
.1869:30 
.186653 
.186377 
.186101 
.185824 
.185548 
.185272 
.184996 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 

9.738048 
.738241 
.738434 
.738627 

3.22 
3.22 
3.22 

391 

9.922768 
.922686 
.922603 
.922520 

1.38 
1.38 
1.38 

9.815280 
.815555 
.815831 
.816107 

4.60 
4.60 
4.59 

A  tQ 

0.184720 
.184445 
.184169 
.183893 

50 
49 
48 
47 

14 
15 
16 
17 

.738820 
.739013 
.739206 
.739398 

3.21 
3.21 
3.21 

.922438 
.922355 
.9222/2 

.922189 

1.38 
1.38 
1.38 

.816382 
.816658 
.816933 
.817209 

4.59 
4.59 
4.59 

.183618 
.183342 
.183067 
.182791 

46 
45 
44 
43 

18 
19 

.739590 
.7397&3 

3.20 
3.20 

.922106 
.922023 

1.38 
1.38 

.817484 
.817759 

4.59 
4.59 

.182516 
.182241 

42 

41 

20 
21 
22 
23 

9.739975 
.740167 
.740359 
.740550 

3.20 
3.20 
3.20 

9.921940 
.921857 
.921774 
.921691 

1.39 
1.39 
1.39 

9.818035 
.818310 
.818585 
.818880 

4.59 
4.58 

4.58 

0.181965 
.181690 
.181415 
.181140 

40 
39 
38 
37 

24 
25 

26 

27 
28 

.740742 
.740934 
.741125 
.741316 
.741508 

3.19 
3.19 
3.19 
3.19 
3.19 

.921607 
.921524 
.921441 
.921357 
.921274 

1.39 
1.39 
1.39 
1.39 

.8191&5 
.819410 
.819684 
.819959 
.820234 

4.58 
4.58 
4.58 
4.58 
4.58 

.180865 
.180590 
.180316 
.180041 
.179766 

36 
35 
34 
33 
32 

29 

.741699 

3.18 

.921190 

1.39 

.820508 

4.58 

.179492 

31 

30 
31 

9.741889 
.742080 

3.18 

9.921107 
.921023 

1.39 

9.820783 
.821057 

4-57 

0.179217 
.178943 

30 
29 

32 

.742271 

31  A 

.920939 

1  40 

.821332 

A  K7 

.178668 

28 

33 
34 
35 
36 
37 
38 
39 

.742402 
.742652 
.742842 
.743033 
.743223 
.743413 
.743602 

3.17 
3.17 
3.17 
3.17 
3.17 
3.16 
3.16 

.920856 
.920772 
.920688 
.920604 
.920520 
.920436 
.920352 

1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 

.821606 

.821880 
.822154 
.822429 
.822703 
.822977 
.823251 

4.57 
4.57 
4.57 
4.57 
4.57 
4.57 
4.56 

.178394 
.178120 
.177846 
.177571 
.177297 
.177023 
.176749 

27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.743792 
.743982 
.744171 
.744361 
.744550 
.744739 
.744928 
.745117 
.745306 
.745494 

3.16 
3.16 
3.16 
3.15 
3.15 
3.15 
3.15 
3.15 
3.14 
3.14 

9.920268 
.920184 
.920099 
.920015 
.919931 
.919846 
.919762 
.919677 
.919593 
.919508 

1.40 
1.40 
1.40 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 

9.823524 
.823798 
.824072 
.824345 
.824619 
.824893 
.825166 
.825439 
.825713 
.825986 

4.56 
4.56 
4.56 
4.56 
4.56 
4.56 
4.56 
4.56 
4.55 
4.55 

0.176476 
.176202 
.175928 
.175655 
.175381 
.175107 
.174834 
.174561 
.174287 
.174014 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

9.745683 
.745871 
.746060 
.746248 
.746436 
.746624 
.746812 
.746999 
.747187 
.747374 

3.14 
3.14 
3.14 
3.13 
3.13 
3.13 
3.13 
3.13 
3.12 

9.919424 
.919339 
.919254 
.919169 
.919085 
.919000 
.918915 
.918830 
.918745 
.918659 

1.41 
1.41 
1.41 
1.41 
1.42 
1.42 
1.42 
1.42 
1.42 

9.826259 
.826532 
.826805 
.827078 
.827351 
.827624 
.827897 
.828170 
.828442 
.828715 

4.55 
4.55 
4.55 
4.55 
4.55 
4.55 
4.55 
4.54 
4.54 

0.173741 
.173468 
.173195 
.172922 
.172649 
.172376 
.172103 
.171830 
.171558 
.171285 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 

60 

.747562 

.918574 

.828987 

.171013 

0 

M. 

Cosine. 

D.  I". 

Sine. 

D.  I". 

Cotang. 

D.  I". 

Tang. 

M. 

133° 


06' 


274          TABLE  XV.   LOGARITHMIC  SINES, 
340                                                  1« 

M. 

Sine 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

1 
2 
3 
4 
5 
6 
7 
8 
9 

9.747562 
.747749 
.747936 
.748123 
.748310 
.748497 
.748683 
.748870 
.749058 
.749243 

3.12 
3.12 

3.12 
3.11 
3.11 
3.11 
3.11 
3.11 
3.10 
3  10 

9.918574 
.918489 
.918404 
.918318 
.918233 
.918147 
.918062 
.917976 
.917891 
.917805 

1.42 
1.42 

1.42 
1.42 
1.42 
143 
1.43 
1.43 
1.43 
1.43 

9.828987 
.829260 
.829532 
.829805 
.830077 
.830349 
.830621 
.830893 
,831165 
.831437 

4.54 
4.54 
4.54 
4.54 
4.54 
4.54 
4.53 
4.53 
4.63 
4.53 

0.171013 
.170740 
.170468 
.170195 
.169923 
.169651 
.169379 
.169107 
.168835 
.168563 

60 
59 
58 
57 
56 
65 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.749429 
.749615 
.749801 
.749987 
.760172 
.760358 
.750543 
.750729 
.750914 
.751099 

3.10 
3.10 
3.10 
3.10 
3.09 
3.09 
3.09 
3.09 
3.09 
308 

9.917719 
.917634 
917548 
.917462 
.917376 
.917290 
917204 
.917118 
917032 
.916946 

1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.44 
1.44 
1  44 

9.831709 
.831981 
.832253 
.832525 
.832796 
.833068 
.833339 
.833611 
.833882 
.834154 

4.53 
4.63 
4.53 
4.53 
4.53 
4.53 
4.52 
4.52 
4.52 
4.52 

0.168291 
.168019 
.167747 
.167476 
.167204 
.166932 
.166661 
.166389 
.166118 
.166846 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
26 
26 
27 
23 
29 

9.751284 
.761469 
.751654 
.751839 
.752023 
.752208 
.752392 
.752576 
752760 
.752944 

3.08 
3.08 
3.08 
3.08 
3.07 
3.07 
3.07 
3.07 
3.07 
3  06 

9.916859 
.916773 
.916687 
.916600 
.916514 
.916427 
.916341 
916254 
916167 
.916081 

1.44 
1.44 
1.44 
1.44 

ft 

1.44 
1.44 
1.45 
1  45 

9.834425 
.834696 
.834967 
.835238 
.835509 
.835780 
.836051 
.836322 
.836593 
.836864 

4.62 
4.52 
4.62 
4.52 
4.52 
4.52 
4.61 
4.51 
4.61 
4.51 

0.165575 
.165304 
.165033 
.164762 
.164491 
.164220 
.163949 
.163678 
.163407 
.163136 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.753128 
.753312 
.753495 
.753679 
.753862 
.754046 
.754229 
.754412 
.754595 
.754778 

3.06 
3.06 
3.06 
3.06 
3.05 
3.05 
3.05 
3.05 
3.05 
305 

9.915994 
.915907 
.915820 
.915733 
.915646 
.915559 
.915472 
.915385 
.915297 
.915210 

1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1  46 

9.837134 

.837405 
.837676 
.837946 
.838216 
.838487 
.838757 
.839027 
.839297 
.839568 

4.51 
4.51 
4.51 
4.51 
4.51 
4.51 
4.60 
4.50 
4.60 
4.60 

0.162860 
.162595 
.162325 
.162054 
.161784 
.161513 
.161243 
.160973 
.160703 
.160432 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 

44 
45 
46 

47 
48 
49 

9.754960 
.755143 
.755326 
.755508 
.755690 
.755872 
.756054 
.756236 
.756418 
.756600 

3.04 
3.04 
3.04 
3.04 
3.04 
3.03 
3.03 
3.03 
3.03 
3  03 

9.915123 
.915035 
.914948 
914860 
.914773 
.914685 
.914598 
.914510 
.914422 
.914334 

1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1  46 

9.839838 
.840108 
.840378 
.840648 
.840917 
.841187 
.841457 
.841727 
.841996 
.842266 

4.50 
4.60 
4.60 
4.60 
4.50 
4.49 
4.49 
4.49 
4.49 
4.49 

0.160162 
.159892 
.159622 
.159352 
.159083 
.158813 
.158543 
.158273 
.158004 
.157734 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
63 
64 
66 
66 
57 
68 
69 
60 

9.756782 
.756963 
.757144 
.757326 
.757507 
.757688 
.757869 
.758050 
.758230 
.758411 
.758591 

3.02 
3.02 
3.02 
3.02 
3.02 
3.02 
3.01 
3.01 
3.01 
3.01 

9.914246 
.914158 
.914070 
.913982 
.913894 
.913806 
.913718 
913630 
.913541 
.913453 
.913365 

1.47 
1.47 
1.47 

1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 

9.842535 

.842805 
.843074 
.843343 
.843612 
.843882 
.844151 
.844420 
.844689 
.844958 
.845227 

4.49 
4.49 
4.49 
4.49 
4.49 
4.49 
4.48 
4.48 
4.48 
4.48 

0.157465 
.157195 
.156920 
.156657 
.156*88 
.156118 
.155849 
.155580 
.155311 
.155042 
.154773 

10 
9 
8 
7 
6 
5 
4 
3 
1 
1 
0 

M. 

Coeirve. 

D.  1". 

Bice. 

D.1" 

Cotang 

D.I*. 

Tang. 

fit 

55° 


850 


COSINES,    TANGENTS,    AND   COTANGENTS. 


M 

Sine. 

D.I* 

Cosine 

D.I' 

Tang. 

D.  1". 

Cotang 

M 

0 

< 

4 
B 

e 

8 
9 

9.758591 
.758772 
.768952 
.759132 
.769312 
.759492 
.769672 
.759862 
.760031 
.760211 

3.01 
3.00 
3.00 
3.00 
3.00 
3.00 
2.99 
2.99 
2.99 
2.99 

9.91336 
.9132? 
.91318 
.91309 
.91301 
.912922 
.912833 
.912744 
.912655 
.912566 

1.47 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 

9.84522 
.84549 
.845764 
.84603 
.846302 
.84657 
.84683 
.847108 
.847376 
.847644 

4.48 
4.48 
4.48 
4.48 
4.48 
4.48 
4.48 
4.47 
4.47 
4  47 

0.15477 
.154504 
.154236 
.15396 
.153698 
153430 
.15316 
.152892 
.152624 
.152356 

60 

81 

67 
56 
66 
64 
63 
52 
51 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9.760390 
.760569 
.760748 
.760927 
.761106 
.761285 
.761464 
.761642 
.761821 
.761999 

2.99 
2.99 
2.98 
2.98 
2.98 
2.98 
2.98 
2.97 
2.97 
2.97 

9.912477 
.912388 
.912299 
.912210 
.912121 
.912031 
.911942 
.911853 
.911763 
.911674 

1.48 
1.48 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 

9.847913 

.84818 
.848449 
.848717 
848986 
.849254 
.849522 
.849790 
.850057 
.850325 

4.47 
4.47 
4.47 
4.47 
447 
4.47 
4.47 
4.46 
4.46 
446 

0.152087 
.161819 
.151651 
.151283 
.151014 
.160746 
.150478 
.150210 
.149943 
.149676 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
21 
22 
23 
24 
26 
26 
27 
28 
29 

9.7C2177 
.762356 
762534 
.762712 
.762889 
.763067 
.763245 
.763422 
.763600 
.763777 

2.97 
2.97 
2.97 
2.96 
2.96 
2.96 
2.96 
2.96 
2.95 
2.96 

9.911584 
.911495 
.911405 
.911315 
911226 
911136 
911046 
910956 
.910866 
.910776 

1.49 
1.49 
1.49 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 

9.850593 
.850861 
851129 
.851396 
.851664 
.851931 
.862199 
.852466 
852733 
.853001 

4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
446 

0.149407 
.149139 
148871 
148604 
.148336 
.148069 
147801 
147534 
.1472*7 
146999 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
84 
36 
36 
37 
38 

3d 

9.763964 
.764131 
764308 
764485 
764662 
.764838 
.765015 
.765191 
.765367 
.765544 

2.95 
2.95 
2.95 
2.95 
2.94 
2.94 
2.94 
2.94 
2.94 
2.P3 

9.910686 
910596 
910506 
910416 
910325 
.910235 
910144 
.910054 
909963 
909873 

1.60 
1.60 
1.60 
1.51 
1.61 
1.61 
1.61 
1.51 
1.61 
1.51 

9.853268 
.853535 
.853802 
.854069 
.854336 
854603 
.854870 
.855137 
.855404 
.866671 

4.46 
4.46 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
444 

0.146732 
.146465 
.146198 
.145931 
-.145664 
.145397 
.145130 
144863 
144596 
.144329 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
46 
46 
47 
48 
49 

9.765720 
.765896 
.766072 
.766247 
.766423 
.766598 
.766774 
.766949 
.767124 
.767300 

2.93 
2.93 
2.93 
2.93 
2.93 
2.92 
2.92 
2.92 
2.92 
2.92 

9.909782 
.909691 
.909601 
.909510 
909419 
909328 
.909237 
.909146 
.909055 
.908964 

1.51 
1.61 
1.61 
1.61 
1.62 
1.52 
1.62 
1  62 
1.62 
1.62 

9.855938 
.856204 
.856471 
.856737 
.857004 
.857270 
.857537 
.857803 
.858069 
.868336 

4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4  44 

0.144062 
.143796 
.143529 
.143263 
.142996 
.142730 
.142463 
.142197 
.141931 
.141664 

20 
19 
18 
17 
16 
15 
14 
3 
2 

60 
61 
62 
63 
64 
66 
66 
67 
68 
69 
60 

9.767476 
.767649 
.767824 
.767999 
.768173 
.768348 
.768522 
.768897 
.768871 
.769045 
.769219 

2.91 
2.91 
2.91 
2.91 
2.91 
2.91 
2.90 
2.90 
2.90 
2.90 

9.908873 
.908781 
.908690 
.908599 
.908507 
.908416 
.908324 
.908233 
.908141 
.908049 
.907958 

1.52 
1.52 
1.52 
1.52 
1.52 
1.53 
1.53 
1.53 
1.53 
1.63 

.868602 
.858868 
.859134 
.859400 
.859666 
.859932 
860198 
860464 
860730 
860995 
861261 

4.44 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 

0.141398 
.141132 
.140866 
.140600 
.140334 
.140068 
.139802 
.139536 
.139270 
.139005 
.138739 

0 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Ooeine. 

D.P. 

Bine. 

.1". 

Cofang. 

D.  1". 

Tang. 

6*0 


TABLE   XV.      LOGARITHMIC   SINES, 


M. 

Bine. 

D.  1". 

Godne. 

D.  1". 

Tang. 

D.  1". 

Ootang. 

M 

0 
1 

3 

6 
6 

8 
9 

9.769219 
.769393 
.769566 
.769740 
.769913 
.770087 
.770260 
.770433 
.770606 
.770779 

2.90 
2.90 
2.89 
2.89 
2.89 
2.89 
2.89 
2.88 
2.88 
288 

.907958 
.907866 
.907774 
.907682 
.907590 
.907498 
.907406 
.907314 
.907222 
.907129 

1.53 
1.53 
1.53 
1.53 
1.53 
1.53 
1.54 
1.54 
1.54 
1.54 

9.861261 
.861527 
.861792 
.862058 
.862323 
.862589 
.862854 
.863119 
.863385 
.863650 

4.43 
4.43 
4.43 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 

0.138739 
.138473 
.138208 
.137942 
.137677 
.137411 
.137146 
.136881 
.136615 
.136350 

60 
59 
58 
57 
56 
56 
64 
63 
5* 
51 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9.770952 
.771125 
.771298 
.771470 
.771643 
.771815 
.771987 
.772159 
.772331 
.772503 

2.88 
2.88 
2.88 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2  86 

9.907037 

.906945 
.906852 
.906760 
.906667 
.906575 
.906482 
.906389 
.906296 
.906204 

1.54 
1.54 
1.54 
1.54 
1.54 
1.64 
1.55 
1.55 
1.55 
155 

9.863915 
.864180 
.864445 
.864710 
.864975 
.865240 
.865505 
.865770 
.866035 
.866300 

4.42 
4.42 
4.42 
4.42 
4.42 
4.41 
4.41 
4.41 
4.41 
4.41 

0.136085 
.135820 
.135555 
.135290 
.135025 
.134760 
.134495 
.134230 
.133965 
.133700 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.772675 
.772847 
.773018 
.773190 
.773361 
.773533 
.773704 
.7738T5 
.774046 
.774217 

2.86 
2.86 
2.86 
2.86 

2.85 
2.85 
2.85 
2.85 
2.85 
285 

9.906111 

.906018 
.905925 
.905832 
.905739 
.905645 
.905552 
.905459 
.905366 
.905272 

1.55 
1.55 
1.55 
1.65 
1.55 
1.55 
1.55 
1.56 
1.56 
1.56 

9.866564 
.866829 
.867094 
.867358 
.867623 
.867887 
.868152 
.868416 
.868680 
.868945 

4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.40 
4.40 

0.133430 
.133171 
.132906 
.132642 
.132377 
.132113 
.131848 
.131584 
.131320 
.131056 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.774388 

.774558 
.774729 
.774899 
.775070 
775240 
.775410 
.775680 
.775750 
.775920 

2.84 
2.84 
2.84 
2.84 
2.84 
2.84 
2.83 
2.83 
2.83 
283 

9.905179 
.905085 
.904992 
.904898 
,904804 
.904711 
.904617 
.904523 
.904429 
.904335 

1.56 
1.56 
1.56 
1.56 
1.56 
1.56 
1.66 
1.67 
1.57 
1  67 

9.869209 
.869473 
.869737 
.870001 
.870265 
.870529 
.870793 
.871057 
.871321 
.871685 

4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 

0.130791 
.130527 
.130263 
.129999 
.129735 
.129471 
.129207 
.128943 
.128679 
.128415 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.776090 
.776259 
.776429 
.776598 
.776768 
.776937 
.777106 
.777275 
.777444 
.777613 

2.83 
2.83 

2.82 
2.82 
2.82 
2.82 
2.82 
2.82 
2.81 
281 

9.904241 
.904147 
.904053 
.903959 
.903864 
.903770 
.903676 
,903581 
.903487 
.903392 

1.57 
1.57 
1.67 
1.57 
157 
1.57 
1.57 
1.67 
1.58 
1.58 

9.871849 
.872112 
.872376 
.872640 
.872903 
.873167 
.873430 
.873694 
.873957 
.874220 

4.40 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 

0.128151 

.127888 
.127624 
.127360 
.127097 
.126833 
.126570 
.126306 
.126043 
.125780 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
61 
52 
53 
54 
65 
66 
67 
68 
69 

9.777781 
.777950 
.778119 

.778287 
.778455 
.778624 
.778792 
.778960 
.779128 
779295 

2.81 
2.81 
2.81 
2.81 
2.80 
2.80 
2.80 
2.80 
2.80 

9.903298 
.903203 
.903108 
.903014 
.902919 
.902824 
,902729 
.902634 
.902539 
,902444 

1.68 
1.58 
1.58 
1.58 
1.58 
1.58 
1.58 
1.58 
1.59 

ICQ 

9.874484 
.874747 
.875010 
.875273 
.875537 
.875800 
.876063 
.876326 
.876589 
.876852 

4.39 
4.39 
4.39 
4.39 
4.38 
4.38 
4.38 
4.33 
4.38 
4  38 

0.125516 
.125253 
.124990 
.124727 
.124463 
.124200 
.123937 
.123674 
.123411 
.123148 

10 
9 
8 
7 
6 
6 
4 
3 
1 
I 

60 

.779463 

2.79 

.902349 

.877114 

.122886 

0 

M 

Ocxdna. 

D.I" 

Sine. 

D.I' 

Ootang. 

D.  1" 

Tang. 

M. 

I860 


COSINES,    TANGENTS,    AND   COTANGENTS. 


277 


M. 

Sine. 

D.1*. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 

9.779463 
.779631 
.779798 

2.79 
2.79 

9.902349 
.902253 
.902158 

1.59 
1.59 

9.877114 
.877377 

.877640 

4.38 
4.38 

0.122886 
.122623 
.122360 

60 
69 

68 

3 
4 

.779966 
.780133 

2.79 
2  79 

.902063 
.901967 

1.59 

.877903 
.878165 

4.38 
4.33 

.122097 
.121835 

57 
66 

6 
6 

7 
8 
9 

.780300 
.780487 
.780634 

.780801 
.780968 

2.78 
2.78 
2.78 
2.78 
2.78 

.901872 
.901776 
.901681 
.901585 
.901490 

1.59 
1.59 
1.59 
1.59 
1.60 

.878428 
.878691 
.878953 
.879216 
.879478 

4.38 
4.38 
4.38 
4.37 
4.37 

.121572 
.121309 
.121047 
.120784 
.120522 

66 
64 
63 
52 
61 

10 

9.781134 

9  7ft 

9.901394 

fin 

9.879741 

0.120259 

50 

11 
12 
13 
14 
15 
16 

.781301 
.781468 
.781634 
.781800 
.781966 
.782132 

2.77 
2.77 
2.77 
2.77 
2.77 

9  77 

.901298 
.901202 
.901106 
.901010 
.900914 
.900818 

1.60 
1.60 
1.60 
1.60 
1.60 
i  An 

.880003 
.880265 
.880528 
.880790 
.881052 
881314 

4.37 
4.37 
4.37 
4.37 
4.37 

.119997 
.119735 
.119472 
.119210 
.118948 
.118686 

49 
48 
47 
46 
45 
44 

17 

18 

.782298 
.782464 

2.76 

9  7fi 

.900722 
.900626 

1.60 
i  fin 

.881577 
.881839 

4.37 

.118423 
.118161 

43 
42 

19 

.782630 

2.76 

.900529 

1.61 

.882101 

4.37 

.117899 

41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.782796 

.782961 
.783127 
.783292 
.783458 
.783623 
.783788 
.783953 
.784118 
.784282 

2.76 
2.76 
2.76 
2.75 
2.75 
2.75 
2.75 
2.75 
2.75 
2.74 

9.900433 
.900337 
.900240 
.900144 
.900047 
.899951 
.899854 
.899767 
.899660 
.899564 

1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.62 

9.882363 
.882625 
.882887 
.883148 
.883410 
.883672 
.883934 
.884196 
.884457 
.884719 

4.37 
4.37 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 

0.117637 
.117376 
.117113 
.116852 
.116590 
.116328 
.116066 
.115804 
.115543 
.115281 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 

9.784447 
.784612 
.784776 
.784941 
.785105 
.785269 

2.74 
2.74 
2.74 
2.74 
2.74 

9  7Q 

9.899467 
.899370 
.899273 
.899176 

.899078 
.898981 

1.62 
1.62 
1.62 
1.62 
1.62 

9.884980 
.885242 
.885504 
.885765 
.886026 
.886288 

4.36 
4.36 
4.36 
4.36 
4.36 

0.115020 
.114758 
.114496 
.114235 
.113974 
.113712 

30 
29 
28 
27 
20 
25 

36 
37 

.785433 
.785597 

2.73 

9  74 

.898884 
.898787 

1.62 

.886549 
.886811 

4.36 
4.36 

.113451 
.113189 

24 
23 

38 
39 

.785761 

.785925 

2.73 
2.73 

.898689 
.898592 

1.62 
1.62 

.887072 
.887333 

4.35 
4.35 
4.35 

.112928 
.112667 

22 
21 

40 
41 
42 
43 
44 
45 
43 
47 
48 
49 

9786089 
.786252 
.786416 
.786579 
.786742 
.786906 
.787069 
.787232 
.787395 
.787557 

2.73 
2.73 
2.72 
2.72 
2.72 
2.72 
2.72 
2.72 
2.71 
2.71 

9.898494 
.898397 
.898299 
.898202 
.898104 
.898006 
.897908 
.897810 
.897712 
.897614 

1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 

9.887594 

.887855 
.888116 
.888378 
.888639 
.888900 
.889161 
.889421 
.889682 
.889943 

4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 

0.112406 
.112145 
.111884 
.111622 
.111361 
.111100 
.110839 
.110579 
.110318 
.110057 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
69 
60 

9.787720 
.787883 
.788045 
.788208 
.788370 
.788532 
.788694 
.788856 
789018 
.789180 
.789342 

2.71 
2.71 
2.71 
2.71 
2.70 
2.70 
2.70 
2.70 
2.70 
2.70 

9.897516 
.897418 
.897320 
.897222 
.897123 
.897025 
.896926 
.896828 
.896729 
.896631 
.896532 

1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 

9.890204 
.890465 
.890725 
.890986 
891247 
.891507 
891768 
.892028 
.892289 
.892549 
.892810 

4.35 
4.35 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 

0.1097% 
.109535 
.109275 
.109014 
.108753 
.108493 
.108232 
.107972 
.107711 
.107451 
.107190 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0 

M. 

Coelno. 

D.l«. 

Sine. 

D.  1". 

Cotaiig 

D.  1". 

Tang. 

M. 

278           TABLE  XV.   LOGARITHMIC  SINES, 
980                                            14  It 

M. 

Sine. 

D.  I". 

Cosine. 

D.I". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 
4 
6 
6 

r 

8 
9 

9.789342 
.789604 
.789665 
.789827 
.789988 
.790149 
.790310 
.790471 
.790632 
.790793 

2.69 
2.69 
2.69 
2.69 
2.69 
2.69 
2.68 
2.68 
2.68 
2.68 

9.896532 
.896433 
.896a35 
.896236 
.896137 
.896038 
.895939 
.895840 
.895741 
.895641 

1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 

9.892810 
.893070 
.893331 
.893591 
.893851 
.894111 
.894372 
.894632 
.894892 
.895152 

4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.33 
4  33 

0.107190 
.106930 
.106669 
106409 
106149 
.105889 
.105628 
.105368 
.105108 
.104848 

60 
59 
68 
67 
66 
55 
54 
63 
52 
51 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9790954 
.791116 
.791276 
.791436 
.791696 
.791767 
.791917 
.792077 
.792237 
.792397 

2.68 
2.68 
2.67 
2.67 
2.67 
2.67 
2.67 
2.67 
2.67 
2.66 

9.895542 
.895443 
.895343 
.895244 
.895145 
.895045 
.894945 
.894846 
.894746 
.894646 

1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 

9.895412 
.895672 
.895932 
.896192 
.896452 
.896712 
.896971 
.897231 
.897491 
.897751 

4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
433 

0.104588 
.104328 
.104068 
.103808 
.103541 
.103288 
.103029 
.102769 
.102509 
.102249 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
26 
26 
27 
28 
29 

9.792567 
.792716 
.792876 
.793035 
.793195 
.793354 
.793514 
.793673 
.793832 
.793991 

2.66 
2.66 
2.66 
2.66 
2.66 
2.66 
2.65 
2.66 
2.65 
2.65 

9.894546 
.894446 
.894346 
.894246 
.894146 
.894046 
.893946 
.893846 
.893745 
.893645 

1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 

9.898010 
.898270 
.898530 
.898789 
.899049 
.899308 
.899568 
.899827 
.900087 
.900346 

4.33 
4.33 
4.33 
4.33 
4.33 
4.32 
4.32 
4.32 
4.32 
432 

0.101990 
.101730 
.101470 
.101211 
.100951 
.100692 
.100432 
.100173 
.099913 
.099664 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
82 
33 
84 
36 
36 
37 
38 
39 

9.794150 
.794308 
.794467 
.794626 
.794784 
.794942 
.795101 
.795259 
.795417 
.795576 

2.65 
2.64 
2.64 
2.64 
2.64 
2.64 
2.64 
2.64 
2.63 
2.63 

9.893544 
.893444 
.893343 
.893243 
.893142 
.893041 
.892940 
.892839 
.892739 
.892638 

1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 

9.900605 
.900864 
.901124 
.901383 
.901642 
.901901 
.902160 
.902420 
.902679 
.902938 

4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
432 

0.099395 
.099136 
.098876 
.098617 
.098358 
.098099 
.097840 
.097580 
.097321 
.097062 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 

44 
46 
46 
47 

48 
49 

9.795733 
.795891 
.796049 
.796206 
.796364 
.796521 
.796679 
.796836 
.796993 
.797150 

2.63 
2.63 
2.63 
2.63 
2.62 
2.62 
2.62 
2.62 
2.62 
2.61 

9.892536 
.892435 
.892334 
.892233 
.892132 
.892030 
.891929 
.891827 
.891726 
.891624 

1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1  69 

9.903197 
.903456 
.903714 
903973 
.904232 
.904491 
.904750 
.905008 
.905267 
.905526 

4.32 
4.32 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4  31 

0.096803 
.096544 
.096286 
.096027 
.095768 
.095509 
.095250 
.094992 
.094733 
.094474 

20 
19 
18 
17 
16 
15 
14 
13 
12 

i. 

60 
61 
62 
63 
64 
66 
66 
67 
68 
69 
60 

9.797307 
.797464 
.797621 
.797777 
.797934 
.798091 
.798247 
.798403 
.798560 
.798716 
.798872 

2.61 
2.61 
2.61 
2.61 
2.61 
2.61 
2.61 
2.60 
2.60 
2.60 

9.891523 
.891421 
.891319 
.891217 
.891115 
.891013 
.890911 
.890809 
.890707 
.890605 
.890503 

1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 

9.905785 
.906043 
.906302 
.906560 
.906819 
.907077 
.907336 
.907594 
.907853 
.908111 
.908369 

4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 

0.094215 
.093957 
.093698 
.093440 
.093181 
.092923 
.092664 
.092406 
.092147 
.091889 
.091631 

10 
9 
8 
7 
6 
6 
4 
3 
2 

0 

M. 

Cosine. 

D.I 

Sine. 

D.  1". 

Cotang, 

D.  1". 

Tang. 

M. 

COSINES,    TANGENTS,    AND   COTANGENTS. 


390 


If. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 
4 
6 
6 
7 
8 
9 

9.798872 
.799028 
.799184 
.799339 
.799495 
.799651 
.799806 
.799962 
.800117 
.800272 

2.60 
2.60 
2.60 
2.59 
2.59 
2.59 
2.59 
2.59 
2.59 
2.69 

9.890503 

.890400 
.890298 
.890195 
.890093 
.889990 
.889888 
.889785 
.889682 
.889579 

1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 

9.908369 
.908628 
.908886 
.909144 
.909402 
.909660 
.909918 
.910177 
.910435 
.910693 

4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 

0.091631 
.091372 
.091114 
.090856 
.090598 
.090340 
.090082 
.089823 
.089565 
.089307 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
ll 
12 
13 
14 
16 
16 
17 
18 
19 

9.800427 
.800582 
.800737 
.800892 
.801047 
.801201 
.801356 
.801511 
.801665 
.801819 

2.63 
2.63 
2.53 
2.53 
2.58 
2.53 
2.57 
2.57 
2.57 
2.57 

9.889477 
.889374 
.889271 
.889168 
.889064 
.888961 
.888853 
.888755 
.888651 
.888548 

1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 

9.910951 
.911209 
.911467 
.911725 
.911982 
.912240 
.912493 
.912756 
.913014 
.913271 

4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 

0.089049 
.088791 
.088533 
.088275 
.088018 
.087760 
.087502 
.087244 
.086986 
.036729 

50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 

9.801973 
.802128 
.802282 
.802436 
.802589 

2.57 
2.57 
2.57 
2.56 

9.888444 
.888341 

.888237 
.888134 
.888030 

.73 
.73 

.73 
.73 

9.913529 

.913787 
.914044 
.914302 
.914560 

4.29 
4.29 
4.29 
4.29 

0.086471 
.086213 
.085956 
.085698 
.085440 

40 
39 
38 
37 
36 

26 
26 

27 
28 
29 

.802743 
.802897 
.803050 
.803204 
.803357 

2.56 
2.56 
2.56 
2.66 
2.56 
2.56 

.887926 
.887822 
.887718 
.887614 
.887510 

.73 
.73 
1.73 
1.73 
1.73 
1.74 

.914817 
.915076 
.915332 
.915590 
.915847 

4.29 
4.29 
4.29 
4.29 
4.29 

.085183 
.084925 
.084663 
.084410 
.084153 

35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 

9.803511 
.803664 
.803817 
.803970 
.804123 
.804276 
.804428 
.804581 
.804734 

2.65 
2.65 
2.55 
2.55 
2.55 
2.55 
2.64 
2.64 

9.837406 
.837302 
.887198 
.887093 

.886989 
.886885 
.886780 
.886676 
.886571 

1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 

9.916104 
.916362 
.916619 
.916877 
.917134 
.917391 
.917648 
.917906 
.918163 

4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 

0.083896 
.083633 
.083381 
.083123 
.082866 
.032609 
.082352 
.082094 
.081837 

30 
29 
28 
27 
26 
25 
24 
23 
22 

39 

.804336 

2.54 
2.64 

.886466 

1.74 
1.76 

.918420 

4.29 

.081580 

21 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

9.805039 
.805191 
.805343 
.805495 
.805647 
.805799 
.805951 
.806103 
.806254 
.806406 

2.54 
2.64 
2.54 
2.53 
2.63 
2.53 
2.53 
2.53 
2.53 
2.52 

9.886362 

.886257 
.886152 
.886047 
.885942 
.885837 
.885732 
.885627 
.885522 
.885416 

1.76 
1.75 
1.76 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.76 

9.918677 
.918934 
.919191 
.919448 
.919705 
.919962 
.920219 
.920476 
.920733 
.920990 

4.23 

4.28 
4.23 
4.28 
4.28 
4.28 
4.23 
4.28 
4.28 
4.23 

0.081323 
.081066 
.080809 
.080552 
.080295 
.080038 
.079781 
.079524 
.079267 
.079010 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
63 
54 
66 
66 
67 
63 
59 
60 

9.806557 
.806709 
.806860 
.807011 
.807163 
.807314 
.807465 
.807615 
.807766 
.807917 
808067 

2.52 
2.52 
2.52 
2.52 
2.52 
2.52 
2.51 
2.51 
2.51 
2.51 

9.885311 

.885205 
.885100 
.384994 
.884889 
.884783 
.884677 
.884572 
.884466 
.884360 
.834254 

1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.77 
1.77 

9.921247 
.921503 
.921760 
.922017 
.922274 
.922530 
.922787 
.923044 
.923300 
.923567 
.923814 

4.28 
4.28 
4.28 

4.28 
4.28 
4.28 
4.23 
4.28 
4.23 
4.23 

0.078753 

.078497 
.078240 
.077983 
.077726 
.077470 
.077213 
.076956 
.076700 
.076443 
.076186 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Casino 

D.  1". 

Slno. 

D.  1". 

Cotang 

D.  1". 

Tang 

M. 

380 


TABLE   XV.       LOGARITHMIC    SINES 


M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.I  . 

Cotang. 

M. 

0 
1 

2 
3 
4 

5 
6 
7 

8 
9 

9.808067 
.808218 
.808368 
.808519 
.808669 
.808819 
808969 
.809119 
.809269 
.809419 

2.61 
2.51 
2.51 
2.50 
2.50 
2.50 
2.50 
2.50 
2.50 
2.50 

9.884254 
.884148 
.884042 
.883936 
.883829 
.883723 
.883617 
.883510 
.883404 
.883297 

1.77 
1.77 
1.77 
1.77 

1.77 
1.77 
1.77 
1.77 

1.78 

1.78 

9.923814 
.924070 
.924327 
.924583 
.924840 
.925096 
.925352 
.925609 
.925865 
.926122 

4.28 
4.28 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
427 

0.076186 
.075930 
.075673 
.075417 
.075160 
.074904 
.074648 
.074391 
.074135 
.073878 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 

9.809569 
.809718 
.809868 
.810017 
.810167 
.810316 
.810465 

2.49 
2.49 
2.49 
2.49 
2.49 
2.49 

9  dfi 

9.883191 

.883084 
.882977 
.882871 
.882764 
.882657 
.882550 

1.78 
1.78 

1.78 
1.78 
1.78 
1.78 

9.926378 
.926634 
.926890 
.927147 
.927403 
.927659 
.927915 

4.27 
4.27 
4.27 
4.27 
4.27 
4.27 

0.073622 
073366 
.073110 
.072853 
.072597 
.072341 
.072085 

50 
49 
48 
47 
46 
45 
44 

17 
18 
19 

.810614 
.810763 
.810912 

2.48 

2.48 
2.48 

.882443 
.882336 
.882229 

1.79 
1.79 
1.79 

.928171 
.928427 

.923684 

4.27 
4.27 
4.27 
4  27 

.071829 
.071573 
.071316 

43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.811061 
.811210 
.811358 
.811507 
.811655 
.811804 
.811952 
.812100 
.812248 
.812396 

2.48 
2.48 
2.48 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 

9.882121 
.882014 
.881907 
.881799 
.881692 
.881584 
.881477 
.881369 
.881261 
.881153 

1.79 
1.79 
1.79 
1.79 
1.79 
1.79 
1.79 
1.80 
1.80 
1.80 

9.928940 
.929196 
.929452 
.929708 
.929964 
.930220 
.930475 
.930731 
.930987 
.931243 

4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.26 
4.26 
4.26 
426 

0.071060 
.070804 
.070548 
.070292 
.070036 
.069780 
.069525 
.069269 
.069013 
.068757 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.812544 
.812692 
.812840 
.812988 
.813135 
.813283 
.813430 
.813578 
.813725 
.813872 

2.46 
2.46 
2.46 
2.46 
2.46 
2.46 
2.46 
2.45 
2.45 
2.45 

9.881046 

.880938 
.880830 
.880722 
.880613 
.880505 
.880397 
.880289 
.880180 
.880072 

1.80 
1.80 
1.80 
1.80 
1.80 
1.80 
1.81 
1.81 
1.81 
1.81 

9.931499 
.931755 
.932010 
.932266 
.932522 
.932778 
.933033 
.933289 
.933545 
.933800 

4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
426 

0.068501 
.068245 
.067990 
.067734 
.067478 
.067222 
.066967 
066711 
.066455 
.066200 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.814019 
.814166 
.814313 
.814460 
.814607 
.814753 
.814900 
.815046 
.815193 
.815339 

2.45 
2.45 
2.45 
2.45 
2.44 
2.44 
2.44 
2.44 
2.44 
2.44 

9.879963 
.879855 
.879746 
.879637 
.879529 
.879420 
.879311 
.879202 
.879093 
.878984 

1.81 
1.81 
1.81 
1.81 
1.81 
1.81 
1.82 
1.82 
1.82 
1.82 

9.934056 
.934311 
.934567 
.934822 
.935078 
.935333 
.935589 
.935844 
.936100 
.936355 

4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
426 

0.065944 
.065689 
.065433 
.065178 
.064922 
.064667 
.06441  1 
.064156 
.063900 
.063645 

20 
19 
18 
17 
16 
15  ' 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
66 
57 
68 
59 
60 

9.815485 
.815632 
.815778 
.815924 
.816069 
.816215 
.816361 
.816507 
.816652 
.816798 
.816943 

2.44 
2.43 
2.43 
2.43 
2.43 
2.43 
2.43 
2.43 
2.42 
2.42 

9.878875 
.878766 
.878656 
.878547 
.878438 
.878328 
.878219 
.878109 
.877999 
.877890 
.877780 

1.82 
1.82 
1.82 
1.82 
1.82 
1.83 
1.83 
1.83 
1.83 
1.83 

9.936611 
.936866 
.937121 
.937377 
.937632 
.937887 
.938142 
.938398 
.938653 
.938908 
.939163 

4.26 
4.26 
4.26 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 

0.063389 
.063134 
.062879 
.062623 
.062368 
.062113 
.061858 
.061602 
.061347 
.061092 
.060837 

10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 

M. 

Cosine 

D.  1". 

Sine 

D.  1". 

Cotang. 

D.I" 

Tang. 

M. 

49- 


COSINES,    TANGENTS,    AND   COTANGENTS. 


281 


M. 

Slue. 

D.1-. 

Cosine. 

D  1". 

Tang. 

D.I*. 

Ootang. 

M. 

0 
1 
2 
3 

4 
6 
6 
7 
8 
9 

9.816943 
.817088 
.817233 
.817379 
.817524 
.817668 
.817813 
.817958 
.818103 
.818247 

2.42 
2.42 
2.42 
2.42 
2.42 
2.41 
2.41 
2.41 
2.41 
2.41 

9.877780 
.877670 
.877560 
.877450 
.877340 
.877230 
.877120 
.877010 
.876899 
.876789 

1.83 
1.83 
1.83 
1.83 
1.84 
1.84 
1.84 
1.84 
1.84 
1.84 

9.939163 
.939418 
.939673 
.939928 
.940183 
.940439 
.940694 
.940949 
.941204 
.941459 

4.26 
4.26 
4.26 
4.25 
4.25 
4.25 
4.26 
4.25 
4.26 
4.26 

0.060837 
.060582 
.060327 
.060072 
.059817 
.059561 
.059306 
.059051 
.058796 
.058541 

60 
69 
68 
67 
66 
65 
54 
53 
52 
61 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.818392 
.818536 
.818681 
.818825 
.818969 
.819113 
.819257 
.819401 
819545 
.819689 

2.41 
2.41 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.39 

9.876678 
.876568 
.876457 
.876347 
.876236 
.876125 
.876014 
.875904 
.875793 
.875682 

1.84 
1.84 
1.84 
1.84 

1.85 
1.85 

1.85 
1.85 
1.85 
1.85 

9.941713 
.941968 
.942223 
.942478 
.942733 
.942988 
.943243 
.943498 
.943752 
.944007 

4.26 
4.26 
4.25 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 

0.058287 
.058032 
.057777 
.057522 
.057267 
.057012 
.056757 
.056502 
.056248 
.055993 

60 
49 
48 
47 
46 
46 
44 
43 
42 
41 

20 

9819832 

9.875571 

9.944262 

0.055738 

40 

21 
22 
23 
I  24 
1  26 
26 
27 
28 

.819976 
.820120 
.820263 
.820406 
.820550 
.820693 
.820836 
.820979 

2^39 
2.39 
2.39 
2.39 
2.39 
2.38 
2.38 

.875459 
.875348 
.875237 
.875126 
.875014 
.874903 
.874791 
.874680 

1.85 
1.85 
1.85 
1.86 
1.86 
1.86 
1.86 
1.86 

.944517 
.944771 
.945026 
.945281 
.945535 
.945790 
.946045 
.946299 

4.26 
4.26 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 

.055483 
.055229 
.054974 
.054719 
.054465 
.054210 
.053955 
.053701 

39 
38 
37 
36 
36 
34 
33 
32 

29 

.821122 

&  33 

.874568 

1.86 
1.86 

.946554 

4.24 
4.24 

.053446 

31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.821265 
.821407 
.821550 
.821693 
.821835 
.821977 
.822120 
.822262 
.822404 
.822546 

2.38 
2.38 
2.38 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 

9.874456 
.874344 
.874232 
.874121 
.874009 
.873896 
.873784 
.873672 
.873560 
.873448 

1.86 

1.86 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 

9.946808 

.947063 
.947318 
.947572 
.947827 
.948081 
.948335 
.948590 
.948844 
.949099 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 

0.053192 
.052937 
.052682 
.052428 
052173 
.051919 
.051665 
.051410 
.051156 
.050901 

30 
29 
28 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 

9.822688 
.822830 
.822972 
.823114 
.823255 
.823397 
.823539 
.823680 

2.37 
2.36 
2.36 
2.36 
2.36 
2.36 
2.36 

9  3A 

9.873335 
.873223 
.873110 

.872998 
.872885 
.872772 
.872659 
.872547 

1.87 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 

9.949353 

.949608 
.949862 
.950116 
.950371 
.950625 
.950879 
.951133 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 

0.060647 
.050392 
.050138 
.049884 
.049629 
.049375 
.049121 
.048867 

20 
19 
18 
17 
16 
15 
14 
13 

48 

.823821 

A.OO 

9  Q£ 

.872434 

1.88 

.951388 

4.24 

.048612 

12 

49 

.823963 

X.0D 

2.35 

.872321 

1.88 
1.88 

.951642 

4.24 
4.24 

.048358 

11 

60 
51 
52 
53 
54 
55 
56 
57 
58 

9.824101 
.824245 
.824386 
.824527 
.824668 
.824808 
.824949 
.825090 
.825230 

2.35 
2.35 
2.35 
2.35 
2.35 
2.34 
2.34 
2.34 

9.872208 
.872095 
.871981 
.871868 
.871755 
.871641 
.871528 
.871414 
.871301 

1.89 
1.89 
1.89 
1.89 

1.89 
1.89 
1.89 
1.89 

9.951896 
.952150 
.952405 
.952659 
.952913 
.953167 
.953421 
.953675 
953929 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.23 

0.048104 
.047850 
.047595 
.047341 
.047087 
.046833 
.046579 
.046325 
.046071 

10 
9 
8 
7 
6 
6 
4 
3 
2 

59 

.825371 

2.34 

.871187 

1.89 

954183 

4.23 

.  045817 

1 

60 

.825511 

2.34 

.871073 

1.90 

.954437 

4.23 

.045563 

0 

M. 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

48° 


282 


TABLE   XV.       LOGARITHMIC   SINES, 


M. 

Sine. 

D.  1''. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9.825511 
.825651 
.825791 
.825931 
.826071 
.826211 
.826351 
.826491 
.826631 
.826770 

2.34 
2.34 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 

9.871073 
.870960 
.870846 
.870732 
.870618 
.870504 
.870390 
.870276 
.870161 
.870047 

1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.91 
1.91 

9.954437 
.954691 
.954946 
.955200 
.955454 
.955708 
.955961 
.956215 
.956469 
.956723 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

0.045563 
.045309 
.045054 
.044800 
.044546 
.044292 
.044039 
.043785 
.043531 
.043277 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

1C 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.826910 
.827049 
.827189 
.827328 
.827467 
.827606 
.827745 
.827884 
.828023 
.828162 

2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.31 
2.31 
231 

9.869933 
.869818 
.869704 
.869589 
.869474 
.869360 
.869245 
.869130 
.869015 
.868900 

1.91 
1.91 
1.91 
1.91 
1.91 
1.91 
1.91 
1.92 
1.92 
1.92 

9.956977 
.957231 
.957485 
.957739 
.957993 
.958247 
958500 
.958754 
.959008 
.959262 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

0.043023 
.042769 
.042515 
.042261 
.042007 
.041753 
.041500 
.041246 
.040992 
.040738 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.828301 
.828439 

.828578 
.828716 
.828855 
.828993 
.829131 
.829269 
.829407 
.829545 

2.31 
2.31 
2.31 
2.31 
2.31 
2.30 
2.30 
2.30 
2.30 
2.30 

9.868785 
.868670 
.868555 
.868440 
.868324 
.868209 
.808093 
.867978 
.867862 
.867747 

1.92 
1.92 
1.92 
1.92 
1.92 
1.92 
1.93 
1.93 
1.93 
1.93 

9.959516 
.959769 
.960023 
.960277 
.960530 
.960784 
.961038 
.961292 
.961545 
.961799 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

0.040484 
.040231 
.039977 
.039723 
.039470 
.039216 
.038962 
.038708 
.038455 
.038201 

40 
39 
38 
37 
36 
33 
34 
33 
33 
31 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

9.829633 
.829821 
.829959 
.830097 
,830234 
830372 
.830509 
.830646 
.830784 
.830921 

2.30 
2.30 
2.29 
2.29 
2.29 
2.29 
2.29 
2.29 
2.29 
229 

9.867631 
.867515 
.867399 
.867283 
.867167 
.867051 
.866935 
.866819 
.866703 
.866586 

1.93 
1.93 
1.93 
1.93 
1.93 
1.94 
1.94 
1.94 
1  94 
1.94 

9.962052 
.962306 
.962560 
.962813 
.963067 
.963320 
.963574 
.963828 
.964081 
.964335 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

0.037948 
.037694 
.037440 
.037187 
.036933 
.036680 
.036426 
.036172 
.035919 
.035665 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.831058 
.831195 
.831332 
.831469 
.831606 
.831742 
.831879 
.832015 
.832152 
.832288 

2.23 

2.28 
2.28 
2.28 
2.28 
2.28 
2.28 
2.27 
2.27 
227 

9.86647u 
.866353 
.866237 
.866120 
.866004 
.865887 
.865770 
.865653 
.865536 
.865419 

1.94 
1.94 
1.94 
1  94 
1.95 
1.95 
1.95 
1.95 
1.95 
1.95 

9.964588 
.964842 
.965095 
.965349 
.965602 
.965855 
.966109 
.966362 
.966616 
.966869 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

0.035412 
.035158 
.034905 
.034651 
.034398 
.034145 
.033891 
.033638 
.033384 
.033131 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
67 
58 
59 
60 

9.832425 
.832561 
.832697 
.832833 
.832969 
.833105 
.833241 
.833377 
.833512 
.8a3648 
.833783 

2.27 
2.27 
2.27 
2.27 
2.27 
2.26 
2.26 
2.26 
2.26 
2.26 

9.865302 
.865185 
.865068 
.864950 
.864833 
.864716 
.864598 
.864481 
.864363 
.864245 
.864127 

1.95 
1.95 
1.95 
1.96 
1.96 
1.96 
1.96 
1.96 
1.96 
1.96 

9.967123 
.967376 
.967629 
.967883 
.963136 
.968389 
.968643 
.968896 
.969149 
.969403 
.969656 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

0.032877 
.032624 
.032371 
.032117 
.031864 
.031611 
.031357 
.031104 
.030851 
.030597 
030344 

10 
9 
S 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Oosiue. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.I'. 

Tang. 

M. 

133° 


430 


COSINES,    TANGENTS,    AND   COTANGENTS. 


M. 

Sine. 

D.  1". 

Cosine. 

D.I" 

Tang. 

D.  1". 

Cotaug. 

M. 

0 
1 
2 
3 

4 
6 
6 

7 
8 
9 

9.833783 
.833919 
.834054 
.834189 
.834325 
.834460 
.834595 
.834730 
.834865 
.834999 

2.26 
2.26 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 

9.864127 
.864010 
.863892 
.863774 
.863656 
.863538 
.863419 
.863301 
.863183 
.863064 

1.96 
1.97 

1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 

9.969656 
.969909 
.970162 
.970416 
.970669 
.970922 
.971175 
.971429 
.971682 
.971935 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
422 

0.030344 
.030091 
.029838 
.029584 
.029331 
.029078 
.028825 
.028571 
.028318 
.028065 

60 
59 
68 
67 
66 
55 
64 
53 
52 
61 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9.835134 
.835269 
.835403 
.835538 
.835672 
.835807 
.835941 
.836075 
.836209 
.836343 

2.24 

2.24 
2.24 
2.24 
2.24 
2.24 
2.24 
2.23 
2.23 
2.23 

9.862946 
.862827 
.862709 
.862590 
.862471 
.862353 
.862234 
.862115 
.861996 
.861877 

1.98 
1.98 
1.98 
1.98 
1.93 
1.98 
1.98 
1.98 
1.98 
1.99 

9.972188 
.972441 
.972695 
.972948 
.973201 
.973454 
.973707 
.973960 
.974213 
.974466 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
422 

0.027812 
.027559 
.027305 
.027052 
.026799 
.026546 
.026293 
.026040 
.025787 
.025534 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
26 
26 
27 
23 
29 

9.836477 
.836611 
.836745 
.836878 
.837012 
.837146 
.837279 
.837412 
.837546 
.837679 

2.23 
2.23 
2.23 
2.23 
2.23 
2.22 
2.22 
2.22 
2.22 
2.22 

9.861758 
.861638 
.861519 
.861400 
.861280 
.861161 
.861041 
.860922 
.860802 
.860682 

1.99 

1.99 
1.99 
1.99 
1.99 
1.99 
1.99 
2.00 
2.00 
2.00 

9.974720 
.974973 
.975226 
.975479 
.975732 
.975985 
.976238 
.976491 
.976744 
.976997 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
422 

0.026280 
.025027 
.024774 
.024521 
.024268 
.024015 
.023762 
.023509 
.023256 
.023003 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
36 
36 
37 
33 
39 

9.837812 
.837945 
.838078 
.838211 
.838344 
.838477 
.838610 
.838742 
.838875 
.839007 

2.22 
2.22 
2.22 
2.21 
2.21 
2.21 
2.21 
2.21 
2.21 
2.21 

9.860562 

.860442 
.860322 
.860202 
.860082 
859962 
.859842 
.859721 
.859601 
.859480 

2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.01 
2.01 
2.01 
201 

9.977250 
.977503 
.977756 
.978009 
.978262 
.978515 
.978768 
.979021 
.979274 
.979527 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
422 

0.022760 
.022497 
.022244 
.021991 
.021738 
.021485 
.021232 
.020979 
.020726 
.020473 

30 
29 
28 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
43 
44 
46 
46 
47 
43 
49 

9.839140 
.839272 
.839404 
.839536 
.839668 
.839800 
.839932 
.840064 
.840196 
.84C328 

2.21 

2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.19 
2.19 

9.859360 
.859239 
.859119 
.858998 
.858877 
.858756 
.858635 
.858514 
.858393 
.858272 

2.01 
2.01 
2.01 
2.01 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 

9.979780 

.980033 
.980286 
.980538 
.980791 
.981044 
.981297 
.981550 
.981803 
.982056 

4.22 
4.22 
4.22 
4.22 
4.22 
4.21 
4.21 
4.21 
4.21 
4.21 

0.020220 
.019967 
.019714 
.019462 
.019209 
.018956 
.018703 
.018450 
.018197 
.017944 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
63 
64 
66 
66 
67 
63 
69 
60 

9.840469 

.840591 
.840722 
.840854 
.840985 
.841116 
.841247 
.841378 
.841509 
.841640 
.841771 

2.19 
2.19 
2.19 
'*  19 
2.19 
2.19 
2.18 
2.18 
2.18 
2.18 

9.858151 

.858029 
.857908 
.857786 
.857665 
.857543 
.857422 
.857300 
.857178 
.857056 
.856934 

2.02 
2.02 
2.02 
2.03 
2.03 
2.03 
2.03 
2.03 
2.03 
2.03 

9.982309 
.982562 
.982814 
.983067 
.983320 
.933573 
.983826 
.984079 
.984332 
.984584 
.984837 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.017691 

.017438 
.017186 
.016933 
.016680 
.016427 
.016174 
.01592! 
.015668 
.015416 
.015163 

10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 

M. 

Oodne. 

D.  1". 

Sloe. 

D.  1". 

Cotaug. 

D.I". 

Tang. 

M. 

284           TABLE  XV.   LOGARITHMIC  SINES, 
440                                                  138" 

M 

Slue. 

D.  1". 

Cosine 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

2 
3 

6 
6 

7 
8 
9 

9.841771 
.841902 
.842033 
.842163 
.842294 
.842424 
.842555 
.842685 
.842815 
.842946 

2.18 
2.18 
2.18 
2.18 
2.17 
2.17 
2.17 
2.17 
2.17 
2.17 

9.856934 
.856312 
.856690 
.856568 
.856446 
.856323 
.856201 
.856078 
.855956 
.855833 

2.03 
2.04 
2.04 
2.04 
2.04 
2.04 
2.04 
2.04 
2.04 
2.04 

9.984837 

.985090 
.985343 
.985596 
.985848 
.986101 
.986354 
.986607. 
.986860 
.987112 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.015163 
.014910 
.014657 
.014404 
.014152 
.013899 
.013646 
.013393 
.013140 
.012888 

60 
69 
68 
67 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9.843076 
.843206 
.843336 
.843468 
.843595 
.843725 
.843855 
.843984 
.844114 
.844243 

2.17 
2.17 
2.16 
2.16 
2.16 
2.16 
2.16 
2.16 
2.16 
2.16 

9.855711 

.855588 
.855465 
.855342 
.855219 
.855096 
.854973 
.854850 
.854727 
.854603 

2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.06 
2.06 

9.987365 
.987618 

.987871 
.988123 
.988376 
.988629 
.988882 
.989134 
.989387 
.989640 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.012635 
.012382 
.012129 
.01  1877 
.011624 
.011371 
.011118 
.010866 
.010613 
.010360 

50 
19 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

9.844372 
.844502 
.844631 
.844760 
.844889 
.845018 
.845147 
.845276 
.845405 
.845533 

2.16 
2.15 
2.15 
2.15 
2.15 
2.15 
2.16 
2.15 
2.14 
2.14 

9.854480 
.854356 
.854233 
.854109 
.853986 
.853862 
.853738 
.853614 
.853490 
.853366 

2.06 
2.06 
2.06 
2.06 
2.06 
2.06 
2.06 
2.07 
2.07 
2.07 

9.989893 
.990145 
.990398 
.990651 
.990903 
.991156 
.991409 
991662 
.991914 
.992167 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.010107 
.009855 
.009602 
.009349 
.009097 
.008844 
.008591 
.008338 
.008086 
.007833 

40 
89 
38 
37 
36 
36 
34 
33 
82 
31 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

9.845662 
.845790 
.845919 
.846047 
.846175 
.846304 
.846432 
.846560 
.846688 
.846816 

2.14 
2.14 
2.14 
2.14 
2.14 
2.14 
2.13 
2.13 
2.13 
2.13 

9.853242 
.853118 
.852994 
.852869 
.852745 
.852620 
.852496 
.852371 
.852247 
.852122 

2.07 
2.07 
2.07 
2.07 
2.07 
2.08 
2.08 
2.08 
2.08 
2.08 

9.992420 
.992672 
.992925 
.993178 
.993431 
.993683 
.993936 
.994189 
.994441 
.994694 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.007580 
.007328 
.007076 
.006822 
.006569 
.006317 
.006064 
.005811 
.005559 
.005306 

30 
29 
23 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

9.846944 
.847071 
.847199 
.847327 
.847454 
.847582 
.847709 
.847836 
.847964 
.848091 

2.13 
2.13 
2.13 
2.13 
2.12 
2.12 
2.12 
2.12 
2.12 
2.12 

9.851997 
.851872 
.851747 
.851622 
.851497 
.851372 
.851246 
.851121 
.850996 
.850870 

2.08 
2.08 
2.08 
2.09 
2.09 
2.09 
2.09 
2.09 
2.09 
2.09 

9.994947 
.995199 
.995452 
.995705 
.995957 
.996210 
.996463 
.996715 
.996968 
.997221 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.006053 
.004801 
.004548 
.004295 
.004043 
.003790 
.003537 
.003285 
.003032 
.002779 

20 
19 
18 
17 
16 
16 
14 
13 
12 
11 

50 
61 
52 
63 
64 
55 
66 
67 
68 
59 
60 

9.848218 
.848345 

.848472 
.848599 
.848726 
.848852 
.848979 
.849106 
.849232 
.849359 
.849485 

2.12 
2.12 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 

9.850745 
.850619 
.850493 
.850368 
.850242 
.850116 
.849990 
.849864 
.849738 
.849611 
.849485 

2.09 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 
2.11 

9.997473 
.997726 
.997979 
.998231 
.998484 
.998737 
.998989 
.999242 
.999495 
.999747 
0.000000 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.002527 
.002274 
.002021 
.001769 
.001516 
.001253 
.001011 
.000758 
.000505 
.000253 
.000000 

10 
9 

8 

6 
6 
4 
3 
2 
1 
0 

M. 

Ooelne. 

D.  1". 

Sine. 

D.F. 

Cotang. 

D.  1". 

Tang. 

M. 

TABLE     XVI. 


NATURAL   SINES  AND  COSINES. 


286 


TABLE   XYI.       NATUKAL    SINES   AND    COSINES. 


~  — 

00 

10 

30 

30 

40 

M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Codn. 

M. 

0 

.00000 

One. 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

«0 

1 

.00029 

One. 

.01774 

.99984 

.03519 

.99938 

.05263 

.99861 

.07005 

.99754 

53 

2 

.00058 

One. 

.01803 

.99984 

.03548 

.99937 

.05292 

.99860 

.07034 

.99752 

58 

3 

.00087 

One. 

.01832 

.99983 

.03577 

.99936 

05321 

.99858 

.07063 

.99750 

37 

4 

.00116 

One. 

.01862 

.99983 

.03606 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

One. 

.01891 

.99982 

.03635 

.99934 

.05379 

.99855 

.07121 

.99746 

56 

6 

.00175 

One. 

.01920 

.99932 

.C8664 

.99933 

.05408 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.01949 

.99981 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742 

53 

8 

.00233 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740 

52 

9 

.00262 

One. 

.02007 

.99980 

.03752 

.99930 

.05495 

.99849 

.07237 

.99738 

51 

10 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349 

.99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99977 

.03368 

.99925 

.05611 

.99842 

.07353 

.99729 

47 

14 

.00407 

.99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

.07411 

.99725 

45 

16 

.00465 

.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838 

.07440 

.99723 

44 

17 

.00495 

.99999 

.0224'> 

,rJ975  .03984 

.99921 

.05727 

.99836 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02269 

.99974  .04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553 

.99998 

.02298 

.99974  .04042 

.99918 

.05785 

.99833 

.07527 

.99716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.05873 

.99827 

.07614 

.99710 

38 

23 

00669 

.99993 

.02414 

.99971 

.04159 

.99913 

.05902 

.99826 

.07643 

.99708 

37 

24 

.00698 

99998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.99819 

.07759 

.99699 

33 

28 

.00814 

.99937 

.02560 

.99967 

.04304 

99907 

.06047 

.99817 

.07788 

.99696 

32 

29 

.00844 

,9(jyy6 

.02589 

.99966 

.04333 

.99906 

.06076 

.99815 

.07817 

.99694 

31 

30 

.00873 

.90996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812 

.07875 

.9968$ 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99902 

.06163 

.99810 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.07933 

.99685 

27 

34 

.00989 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

.07962 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680 

25 

36 

.01047 

.99995 

.02792 

.99961 

.04536 

.99897 

.06279 

.99803 

.03020 

.99678 

24 

37 

.01076 

.99994 

.02821 

.99960 

.04565 

.99896 

.06308 

.99801 

.08049 

.99676 

23 

38 

.01105 

.99994 

.02350 

.99959 

.04594 

.99894 

.06337 

.99799 

.08078 

.99673 

22 

39 

.01134 

.99994 

.02879 

.99959 

.04623 

.99893 

.06366 

.99797 

.08107 

.99671 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99388 

.06482 

.99790 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

.99886 

06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

99657 

15 

46 

.01338 

.99991 

03083 

.99952 

.04827 

.99883 

.06569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04356 

.99882 

.06598 

.99782 

.08330 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.03363 

.99649 

12 

49 

.01425 

.99990 

.03170 

.99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

.99939 

.03199 

.99949 

.04943 

.99378 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483 

.99939 

.03223 

-99948 

.04972 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513 

.99989 

.03257 

.99947 

.05001 

.99875 

.06743 

.99772 

.08484 

.99639 

8 

53 

.01542 

.99988 

03236 

.99946 

.05030 

.99873 

.06773 

.99770 

.08513 

.99637 

7 

54 

.01571 

.99938 

.03316 

.99945 

.05059 

.99872 

.06802 

.99768 

.08542 

.99635 

6 

55 

.01600 

.99987 

03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.03571 

.99632 

5 

56 

.01629 

.99937 

.03374 

.99943 

.05117 

.99869 

.06360 

.99764 

.08600 

.99630 

4 

57 

.01658 

.99986 

.03403 

.99942 

.05146 

.99867 

.06389 

.99762 

.08629 

.99627 

3 

58 

.01637 

.99936 

.03432 

.99941 

.05175 

.99866 

.06918 

.99760 

.08658 

.99625 

2 

59 

.01716 

.99933 

.03461 

.99940 

.05205 

.99364 

.06947 

.99758 

.08687 

.99622 

1 

60 

.01745 

.99935 

.03490 

.99939 

.05234 

.99363 

.06976 

.99756 

.08716 

.99619 

0 

M. 

Cosin. 

Slue 

Cosin. 

Sine. 

Cosln. 

Sine. 

Cosin. 

Sine. 

Coflln. 

Sine. 

M. 

890 

880 

87° 

860 

850 

TABLE   XVI.      NATURAL   SINES   AND   COSINES. 


287 


50 

GO 

70 

80 

90 

M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

M. 

0 

.08716 

.99619 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15642 

.98769 

60 

1 

.08745 

.99617 

.10482 

.99419 

.12216 

.99251 

.13946 

.99023 

.15672 

.98764 

59 

2 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701 

.93760 

58 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755 

57 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.99240 

.14033 

.99011 

.15758 

.98751 

59 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

.15787 

.98746 

56 

6 

.08889 

.99604 

.10626 

.99434 

.12360 

.99233 

.14090 

.99002 

.15816 

.98741 

54 

7 

.08918 

.99602 

.10655 

.99431 

.12389 

.99230 

.14119 

,98993 

.15845 

.98737 

53 

8 

.08947 

.99599 

.10684 

.99428 

.12418 

.99226 

.14148 

.93994 

.15873 

.98732 

52 

9 

.08976 

.99596 

.10713 

.99424 

.12447 

'.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98932 

.15959 

.98718 

49 

12 

.09063 

.99583 

.10800 

.99415 

.12533 

.99211 

.14263 

.98978 

.15988 

.98714 

48 

13 

.09092 

.99586 

.10329 

.99412 

.12562 

.99203 

.14292 

.98973 

.16017 

.98709 

47 

14 

.09121 

.99583 

.10858 

.99409 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704 

46 

15 

.09150 

.99580 

.10387 

.99406 

.12620 

.99200 

.14349 

.98965 

.16074 

.98700 

45 

16 

.09179 

.99578 

.10916 

.99402 

.12649 

.99197 

.14378 

.98961 

.16103 

.98695 

44 

17 

.09208 

.99575 

.10945 

.99399 

.12678 

.99193 

.14407 

.98957 

.16132 

.98690 

43 

18 

.09237 

.99572 

.10973 

.99396 

.12706 

.99189 

.14436 

.98953 

.16160 

.98686 

42 

19 

.09266 

.99570 

.11002 

.99393 

.12735 

.99186 

.14464 

.98948 

.16189 

.98631 

41 

20 

.09295 

.99567 

.11031 

.99390 

.12764 

.99182 

.14493 

.98944 

.16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

.993S6 

.12793 

.99178 

.14522 

.98940 

.16246 

.98671 

39 

22 

.09353 

.99562 

.11089 

.99383 

.12822 

.99175 

.14551 

.98936 

.16275 

.98667 

38 

23 

.09382 

•99559 

.11118 

.99330 

.12851 

.99171 

.14580 

.98931 

.16304 

.98662 

37 

24 

.09411 

.99556 

.11147 

.99377 

.12880 

.99167 

.14608 

.98927 

.16333 

.98667 

36 

25 

.09440 

.99553 

.11176 

.99374 

.12903 

.99163 

.14637 

.98923 

.16361 

.98652 

35 

26 

.09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.  14666 

.98919 

.16390 

.98648 

34 

27 

.09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695 

.98914 

.16419 

.98643 

33 

28 

.09527  .99545 

.11263 

.99364 

.12995 

.99152 

.  14723 

.98910 

.16447 

.98638 

32 

39 

.09556 

.99542 

.11291 

.99360 

.13024 

.99148 

.14752 

.98906 

.16476 

.93633 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

.16505 

.98629 

30 

31 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137 

.14838 

.98893 

.16562 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.99344 

.13163 

.99129 

.14896 

.98884 

.16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98330 

.16648 

.98604 

25 

36 

.09758 

.99523 

.11494 

.99337 

.13226 

.99122 

.14954 

.98876 

.16677 

.98600 

24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

.16706 

.98595 

23 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.98590 

22 

39 

.09845 

.99514 

.11580 

.99327 

.13312 

.99110 

.15040 

.98863 

.16763 

.98585 

21 

40 

.09874 

.99511 

.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

.99102 

.15097 

.98854 

.16820 

.98575 

19 

42 

.09932 

.99506 

.11667 

.99317 

.13399 

.99098 

.15126 

.98849 

16849 

.98570 

18 

43 

.09961 

.99503 

.11696 

.99314 

.13427 

.99094 

.15155 

.98845 

.16878 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99310 

.13456 

.99091 

.15184 

.98841 

.16906 

.98561 

16 

45 

.10019 

.99497 

.11754 

.99307 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.15241 

.98832 

.16964 

.98551 

14 

47 

.10077 

.99491 

.11812 

.99300 

.13543 

.99079 

.15270 

.98827 

.16992 

.98546 

13 

48 

.10106 

.99438 

.11840 

.99297 

.13572 

.99075 

.15299 

.98823 

.17021 

.98541 

12 

49 

.10135 

.99485 

.11869 

.99293 

.13600 

.99071 

.15327 

.98818 

.17050 

.98536 

11 

50 

.10164 

.99482 

.11893 

.99290 

.13629 

.99067 

.15356 

.98814 

.17078 

.98531 

10 

51 

.10192 

.99479 

.11927 

.99286 

.13658 

.99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

.11956 

.99283 

.13687 

.99059 

.15414 

.98805 

.17136 

.98521 

8 

53 

.10250 

.99473 

.11985 

.99279 

.13716 

.99055 

.15442 

.98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

.99051 

.15471 

.98796 

.17193 

.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

15500 

.98791 

.17222 

.98506 

5 

56 

.10337 

.99464 

.12071 

.99269 

.13802 

.99043 

.15529 

.93787 

.17250 

.98501 

4 

57 

.10366 

.99461 

.12100 

.99265 

.13331 

.99039 

.15557 

.98782 

.17279 

.98496 

3 

58 

.10395 

.99458 

.12129 

.99262 

.13860 

.99035 

.15586 

.98778 

.17308 

.98491 

2 

59 

.10424 

.99455 

.12158 

.99258 

.13889 

.99031 

.15615 

.98773 

.17336 

.98486 

1 

60 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

.17365 

.98481 

0 

M7 

Cosln. 

Sine. 

CoainJ 

Sine. 

CoHtn. 

Sine. 

Cosin. 

Sine. 

Cosln. 

Sine. 

M. 

8*0 

83° 

83° 

810 

800 

288 


TABLE    XVI.       NATURAL    SINES    AND    COSINES. 


100   , 

110 

130 

13° 

14° 

M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine,  i 

Cosin. 

Sine. 

Cosin. 

Sine. 

Corfn. 

M.  ! 

0 

17365 

.98481 

.19081 

98163 

.20791 

.97815 

"32495 

.97437 

.24192 

.97030 

60 

17393 

.98476 

.19109 

98157 

.20820 

.97809 

.22523 

.97430 

.24220 

97023 

59 

2 

17422 

.98471 

.19138 

98152 

.20848 

.97803 

.22552 

.97424 

.24249 

.97015 

58 

3 

17451 

.98466 

.19167 

.98146 

.20877 

.97797 

.22580 

.97417 

.24277 

.97008 

57 

4 

17479 

.98461 

.19195 

.98140 

.20905 

.97791 

.22608 

.97411 

.24305 

.97001 

56 

5 

17508 

.98455 

.19224 

.98135 

.20933 

.97784 

.22637 

.97404 

.24333 

.96994 

55 

6 

17537 

.98450 

.  19252 

93129 

.20962 

.97778 

.22665 

.97398 

.24362 

.96987 

54  I 

7 

17565 

.98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.96980 

53 

8 

17594 

.98440 

.19309 

.98118 

.21019 

.97766 

.22722 

.97384 

.24418 

.96973 

52 

9 

17623 

.98435 

.19338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10 

17651 

.98430 

.19366 

.98107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

17680 

.98425 

.19395 

.98101 

.21104 

.97748 

.22807 

.97365 

.24503 

.96952 

49 

12 

.17708 

.98420 

.19423 

.98096 

.21132 

.97742 

.22835 

.97358 

.24531 

.96945 

48  1 

13 

17737 

.98414 

.19452 

.98090 

.21161 

.97735 

.22863 

.97351 

.24559 

.96937 

47  1 

14 

.17766 

.98409 

.19481 

.98084 

.21189 

.97729 

.22892 

97345 

.24587 

.96930 

46  1 

15 

.17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920 

.97338 

.24615 

.96923 

45  l 

16 

.17823 

.98399 

.19538 

.98073 

.2i246 

.97717 

.22948 

.97331 

.24644 

.96916 

44 

17 

.17852 

.98394 

.19566 

.98067 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

18 

.17880 

.98389 

.19595 

.98061 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19 

.17909 

.98383 

.19623 

.98056 

.21331 

.97698 

23033 

.97311 

.24728 

.96894 

41 

20 

.17937 

.98378 

19652 

.98050 

.21360 

.97692 

.23062 

.97304 

.24756 

.96887 

40 

21 

.17966 

.98373 

.19680 

.98044 

.21388 

.97686 

.23090 

.97298 

.24784 

.96880 

39 

22 

.17995 

.98368 

.19709 

.98039 

.21417 

.97680 

.23118 

.97291 

.24813 

.96873 

38 

23 

.18023 

.98362 

.19737 

.98033 

.21445 

.97673 

.23146 

.97284 

.24841 

.96866 

37 

24 

.18052 

.98357 

.19766 

.98027 

.21474 

.97667 

.23176 

.97278 

.24869 

.96858 

36 

25 

.18081 

.98352 

.19794 

.98021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109 

.98347 

.19823 

.98016 

.21530 

.97655 

.23231 

.97264 

.24925 

.96844 

34 

27 

.18133 

.98341 

.19851 

.98010 

.21559 

.97648 

.23260 

.97257 

.24954 

.96837 

33 

23 

.18166 

.98336 

.19880 

.98004 

.21587 

.97642 

.23288 

.97251 

.24982 

.96829 

32 

29 

.18195 

.98331 

.19908 

.97998 

.21616 

.97636 

.23316 

.97244 

.25010 

.96822 

31 

30 

.18224 

.98325 

.19937 

.97992 

.21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

.97987 

.21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

.18281 

.98315 

.  19994 

.97981 

.21701 

.97617 

.23401 

.97223 

.25094 

.96800 

28 

33 

.18309 

.98310 

.20022 

.97975 

.21729 

.97611 

.23429 

.97217 

.25122 

.96793 

27 

34 

.18338 

.98304 

.20051 

.97969 

.21738 

.97604 

.23458 

.97210 

.25151 

.96786 

26 

35 

.18367 

.98299 

.20079 

.97963 

.21786 

.97598 

.23486 

.97203 

.25179 

.96778 

25 

36 

.18395 

.98294 

.20108 

.97958 

.21814 

.97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

.18424 

.98288 

.20136 

.97952 

.21843 

.97585 

.23542 

.97189 

.25235 

.96764 

23 

38 

.18452 

.98283 

.20165 

.97946 

.21871 

.97579 

.23571 

.97182 

.25263 

.96756 

22 

39 

.18481 

.98277 

.20193 

.97940 

.21899 

.97573 

.23599 

.97176 

.25291 

.96749 

21 

40 

.18509 

.98272 

.20222 

.97934 

.21928 

.97566 

.23627 

.97169 

.25320 

.96742 

20 

41 

.18538 

.98267 

.20250 

.97928 

.21956 

.97560 

.23656 

.97162 

.25343 

.96734 

19 

42 

.18567 

.98261 

.20279 

.97922 

.21985 

.97553 

.23684 

.97155 

.25376 

.96727 

18 

43 

.18595 

.98256 

20307 

.97916 

.22013 

.97547 

.23712 

.97148 

.25404 

.96719 

17 

44 

.18624 

.98250 

.20336 

.97910 

.22041 

.97541 

.23740 

.97141 

.25432 

.96712 

16 

45 

.18652 

.98245 

20364 

.97905 

.22070 

.97534 

.23769 

.97134 

.25460 

.96705 

15 

46 

.18681 

.98240 

20393 

.97899 

.22098 

.97528 

.23797 

.97127 

.25488 

.96697 

14 

47 

.18710 

.98234 

.20421 

.97893 

.22126 

.97521 

.23825 

.97120 

.25516 

.96690 

13 

48 

.18738 

.98229 

.20450 

.97887 

.22155 

.97515 

.23853 

.97113 

.25545 

.96682 

12 

49 

.18767 

.98223 

.20478 

.97881 

.22183 

.97508 

23882 

.97106 

.25573 

.96675 

11 

50 

.18795 

.93218 

.20507 

.97875 

.22212 

.97502 

.23910 

.97100 

.25601 

.96667 

10 

51 

.18824 

.98212 

.20535 

.97869 

.22240 

.97496 

.23938 

.97093 

.25629 

.96660 

9 

52 

.18852 

,98207 

.20563 

.97863 

.22268 

.97489 

.23966 

.97086 

.25657 

.96653 

8 

53 

.18881 

.98201 

.20592 

.97857 

.22297 

.97483 

.23995 

.97079 

.25685 

.96645 

7 

54 

.18910 

.98196 

.20620 

.97851 

.22325 

.97476 

.24023 

.97072 

.25713 

.96638 

6 

55 

18938 

.98190 

.20649 

.97845 

.22353 

.97470 

.24051 

.97065 

.25741 

.96630 

5 

56 

.18967 

.98185 

.20677 

.97839 

.22382 

.97463 

.24079 

.97058 

.25769 

.96623 

4 

57 

.18995 

.98179 

.20706 

.97833 

.22410 

.97457 

.24108 

.97051 

.25798 

.96615 

3 

58 

.19024 

.98174 

.20734 

.97827 

.22438 

.97450 

.24136 

.97044 

.25826 

.96608 

2 

59 

.19052 

.98168 

.20763 

.97821 

.22467 

.97444 

.24164 

.97037 

.25854 

.96600 

1 

60 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.25882 

96593 

0 

M. 

Coein. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosir  . 

Sine. 

Cosin. 

Sine. 

M. 

790 

yso 

77o 

760 

7»o 

.,.-  i 

TABLE   XVI.       NATURAL   SINES   AND   COSINES. 


289 


150 

160 

170 

180 

190 

M 

Sine. 

Cosin 

Sine. 

Cosin 

Sine. 

Cosin 

Sine. 

Cosin. 

Sine. 

Ciwin. 

M. 

0 

.25882 

.96593 

.27564 

.9~6f26 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

60 

1 

.25910 

.96535 

.27592 

.96118 

.29265 

.95622 

.30929 

.95097 

.32584 

.94542 

59 

4 

.25933 

.96573 

.27620 

.96110 

.29293 

.95613 

.30957 

.95088 

.32612 

.94533 

58 

3 

.25966 

.96570 

.27643 

.96102 

.29321 

95605 

.30985 

.95079 

.32639 

.94523 

57 

4 

.25994 

.96562 

.27676 

.96094 

.29348 

.95596 

.31012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.96555 

.27704 

.96086 

.29376 

.95538 

.31040 

.95061 

.32694 

.94504 

55 

6 

.26050 

.96547 

.27731 

.96073 

.29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7 

.26079 

.96540 

.27759 

.96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94435 

53 

8 

.26107 

.96532 

.27787 

.96062 

.29460 

.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135 

.96524 

.27815 

.96054 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466 

51 

10 

26163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

.95015 

.32332 

.94457 

50 

11 

26191 

.96509 

.27871 

.96037 

.29543 

.95536 

.31206 

.95006 

.32859 

.94447 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.9552S 

.31233 

.94997 

.32887 

.94438 

48 

13 

.26247 

.96-194 

.27927 

.96021 

.29599 

.95519 

.31261 

.94988 

.32914 

.94428 

47 

14 

.26275 

.96436 

.27955 

.96013 

.29626 

.95511 

.31289 

.94979 

.32942 

.94418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31316 

.94970 

.32969 

.94409 

45 

16 

.26331 

.96471 

.28011 

.95997 

.29682 

.95493 

.31344 

.94961 

.32997 

.94399 

44 

17 

.26359 

.96463 

.28039 

.95939 

.29710 

.95485 

.31372 

.94952 

.33024 

.94390 

43 

13 

.26337 

.96456 

.23067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.26415 

96448 

.23095 

.95972 

.29765 

.95467 

.31427 

.94933 

.33079 

.94370 

41 

20 

.26443 

.96440 

.23123 

.95964 

.29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40 

21 

.26471 

.96433 

.23150 

.95956 

.29821 

.95450 

.31482 

.94915 

.33134 

.94351 

39 

22 

.26500 

.96425 

.28178 

.95943 

.29849 

.95441 

.31510 

.94906 

.33161 

.94342 

38 

23 

.26523 

.96417 

.28206 

.95940 

.29876 

.95433 

.31537 

.94897 

.33189 

.94332 

37 

1  24 

.26556 

.96410 

.28234 

.95931 

.29904 

95424 

.31565 

.94888 

.33216 

.94322 

36 

25 

26534 

.96402 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313 

35 

26 

.26612 

.96394 

.28290 

.95915 

.29960 

.95407 

.31620 

.94869 

.33271 

.94303 

34 

27 

.26640 

.963S6 

.2-3318 

.95907 

.29987 

.95398 

.31648 

.94860 

.33298 

.94293 

33 

28 

.26663 

.96379 

.28346 

.95898 

.30015 

.95aS9 

.31675 

.94851 

.33326 

.94284 

32 

29 

.26696 

.96371 

.28374 

.95890 

.30043 

.95380 

.31703 

.94842 

.33353 

.94274 

31 

30 

.26724 

.96363 

.28402 

.95882 

.30071 

.95372 

.31730 

.94832 

.33381 

.94264 

30 

31 

26752 

.96355 

.28429 

.95874 

.30098 

.95363 

.31758 

.94823 

.33408 

.94254 

29 

32 

.26780 

.96347 

.28457 

.95865 

.30126 

.95354 

.31786 

.94814 

.33436 

.94245 

28 

33 

26808 

.96340 

.28485 

.95357 

.30154 

.95345 

.31813 

.94805 

.33463 

.94235 

27 

34 

26836  .96332 

.28513 

.95349 

.30182 

.95337 

.31841 

.94795 

.33490 

.94225 

26 

35 

26864  1.96324 

.28541 

.95841 

.30209 

95323 

.31868 

.94786 

.33518 

.94215 

25 

36 

26892  .96316 

.28569 

95332 

.30237 

95319 

.31896 

.94777 

.33545 

.94206 

24 

37 

269201.96303 

.23597 

95324 

.30265 

95310 

.31923 

.94763 

.33573 

.94196 

23 

38 

26948 

.96301 

.28625 

95316 

.30292 

95301 

.31951 

.94758 

.33600 

.94186 

22 

39 

26976 

.96293 

28652 

95807 

.30320 

95293 

.31979 

94749 

.33627 

.94176 

21 

40 

27004 

.96235 

.23680 

95799 

.30348 

95234 

.32006 

.94740 

.33655 

.94167 

20 

41 

27032 

.96277 

.28708 

95791 

30376 

95275 

.32034 

94730 

.33682 

.94157 

19 

42 

27060 

.96269 

28736 

95782 

30403 

95266 

.32061 

94721 

.33710 

.94147 

18 

43 

27088 

.96261 

28764 

95774 

30431 

95257 

.32089 

94712 

.33737 

.94137 

17 

44 

27116 

.96253 

28792 

95766 

30459 

95248 

32116 

94702 

.33764 

.94127 

16 

45 

27144 

.96246 

28820 

95757 

30486 

95240 

32144 

94693 

.33792 

.94118 

15 

46 

27172 

.96238 

28847 

95749 

30514 

95231 

32171 

94684 

.33819 

.94108 

14 

47 

27200 

.96230 

28875 

95740 

30542 

95222 

32199 

94674 

.33846 

.94098 

13 

48 

27228 

.96222 

23903 

95732 

30570 

95213 

32227 

94665 

.33874 

.94088 

12 

49 

27256 

.96214 

28931 

957^ 

30597 

95204 

32254 

94656 

.33901 

.94078 

11 

50 

27234 

.96206 

28959 

95716 

30625 

95195 

32232 

94646 

.33929 

.94068 

10 

61 

27312 

.96193 

28987 

95707 

30653 

95136 

32309 

94637 

.33956 

.94058 

9 

52 

27340 

.96,90 

29015 

95693 

30680 

95177 

32337 

94627 

33983 

.94049 

8 

53 

27363 

.96182 

29042 

95690 

30708 

95163 

32364 

94618 

.34011 

.94039 

7 

54  27396 

.96174 

29070 

95631 

30736 

95159 

32392 

94609 

34038 

.94029 

6 

55 

.27424 

.96166 

29093 

95673 

30763 

95150 

32419 

94599 

34065 

.94019 

5 

56 

.27452  .96153 

29126;  95664 

30791 

.95142 

32447 

94590 

34093 

.94009 

4 

57 

.27480  .96150 

29M4  95656 

30319  .95133 

32474 

94530 

.34120 

.93999 

3 

58 

.27503  .96142 

29  1  32  .95647 

30346  .95124 

32502 

94571 

.34147 

.93989 

2 

59 

.27536  .96131 

29200  .95639 

30874  >.951  15 

32529 

94561 

34175 

.93979 

60 

.27564  96126 

29237  95630 

309021.95106 

32557 

94552 

34202 

.93969 

0 

M. 

Coda.  Sine. 

Cosin.  Sine. 

Cosin.  Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

»: 



740 

730       730    |    710 

700 

20 


290 


TABLE    XVI.       NATURAL    SINES    AND    COSINES. 


30° 

310 

33° 

330 

940 

1 

M 

Sine. 

Cosin 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Corfn. 

M. 

0 

.34202 

.93969 

.35337 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91355 

60 

.34229 

.93959 

.35864 

.93348 

.37488 

.92707 

.39100 

.92039 

.40700 

.91343 

59 

2 

.34257 

.93949 

.35391 

.93337 

.37515 

.92697 

.39127 

.92028 

.40727 

.91331 

58 

3 

.34284 

.93939 

.35913 

.93327 

.37542 

.92686 

.39153 

.92016 

.40753 

.91319 

67 

4 

34311 

.93929 

.35945 

.93316 

.37569 

.92675 

.39180 

.92005 

.40780 

.91307 

66 

5 

.34339 

.93919 

.35973 

.93306 

.37595 

.92664 

.39207 

.91994 

.40806 

.91295 

55 

6 

.34366 

.93909 

.36000 

.93295 

.37622 

.92653 

.39234 

.91982 

.40833 

.91283 

54 

7 

.34393 

.93899, 

.36027 

.93285 

.37649 

.92642 

.39260 

.91971 

.40860 

.91272 

53 

8 

.34421 

.93889 

.36054 

.93274 

.37676 

.92631 

.39237 

.91959 

.40886 

.91260 

52 

9 

.34448 

.93879 

.36081 

.93264 

.37703 

.92620 

.39314 

.91948 

.40913 

.91248 

51 

10 

.34475 

.93369 

.36108 

.93253 

.37730 

.92609 

.39341 

.91936 

.40939 

.91236 

50 

11 

.34503 

.93859 

.36135 

.93243 

.37757 

.92598 

.39367 

.91925 

.40966 

.91224 

49 

12 

.34530 

.93849 

.36162 

.93232 

.37784 

.92587 

.39394 

.91914 

.40992 

.91212 

48 

13 

.34557 

.93339 

.36190 

.93222 

.37811 

.92576 

.39421 

.91902 

.41019 

.91200 

47 

14 

.34584 

.93829 

.36217 

.93211 

.37838 

.92565 

.39448 

.91891 

.41045 

,91188 

46 

15 

.34612 

.93819 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

45 

16 

.34639 

.93309 

.36271 

.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91164 

44 

17 

.34666 

.93799 

.36298 

.93180 

.37919 

.92532 

.39528 

.91856 

.41125 

.91152 

43 

18 

.34694 

.93789 

.36325 

.93169 

.37946 

.92521 

.39555 

.91845 

.41151 

.91140 

42 

;  19 

.34721 

.93779 

.36352 

.93159 

.37973 

.92510 

.39581 

.91833 

.41178 

.91128 

41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116 

40 

21 

.34775 

.93759 

.36406 

.93137 

.38026 

.92488 

.39635 

.91810 

.41231 

.91104 

39 

22 

.34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092 

38 

23 

.34830 

.93738 

.36461 

.93116 

.33080 

.92466 

.39688 

.91787 

.41284 

.91080 

37 

24 

.34857 

.93728 

.36488 

.93106 

.38107 

.92455 

.39715 

.91775 

.41310 

.91068 

36 

25 

.34884 

.93718 

.36515 

.93095 

.38134 

.92444 

.39741 

.91764 

.41337 

.91056 

35 

26 

.34912 

.93703 

.36542 

.93084 

.38161 

.92432 

.39768 

.91752 

.41363 

.91044 

34 

27 

.34939 

.93693 

.36569 

.93074 

.33188 

.92421 

.39795 

.91741 

.41390 

.91032 

33 

28 

.34966 

.93688 

.36596 

.93063 

.38215 

.92410 

.39822 

.91729 

.41416 

.91020 

32 

29 

.34993 

.93677 

.36023 

.93052 

.38241 

.92399 

.39848 

.91718 

.41443 

.91008 

31 

30 

.36021 

.93667 

.36650 

.93042 

.38263 

.92388 

.39376 

.91706 

.41469 

.90996 

30 

31 

.35048 

.93657 

.36677 

.93031 

.33295 

.92377 

.39908 

.91694 

.41496 

.90984 

29 

32 

.35075 

.93647 

.36704 

.93020 

.38322 

.92366 

.39928 

.91683 

.41522 

.90972 

28 

33 

.35102 

.93637 

.36731 

.93010 

.33349 

.92355 

.39955 

.91671 

.41549 

.90960 

27 

34 

.35130 

.93626 

.36758 

.92999 

.38376 

.92343 

.39982 

.91660 

.41575 

.90948 

26 

35 

.85157 

.93616 

.36785 

.92983 

.33403 

.92332 

.40008 

.91648 

.41602 

.90936 

25 

36 

.35184 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91636 

.41628 

.90924 

24 

37 

.35211 

.93596 

.36339 

.92967 

.38456 

.92310 

.40062 

.91625 

.41655 

.90911 

23 

38 

.35239 

.93585 

.36367 

.92956 

.33483 

.92299 

.40088 

.91613 

.41681 

.90899 

22 

39 

.35266 

.93575 

.36894 

.92945 

.33510 

.92287 

.40115 

.91601 

.41707 

.90887 

21 

40 

.35293 

.93565 

.36921 

.92935 

.38537 

.92276 

.40141 

.91590 

.41734 

.90875 

20 

41 

.35320 

.93555 

.36943 

.92924 

.38564 

.92265 

.40168 

.91578 

.41760 

.90863 

19 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.92254 

.40195 

.91566 

.41787 

.90861 

18 

43 

.35375 

.93534 

.37002 

.92902 

.38617 

.92243 

.40221 

.91555 

.41813 

.90839 

17 

44 

.35402 

.93524 

.37029 

.92892 

.38644 

.92231 

.40248 

.911543 

.41840 

.90826 

16 

45 

.35429 

.93514 

.37056 

.92831 

.38671 

.92220 

.40275 

.91531 

.41866 

.90814 

15 

46 

.35456 

.93503 

.37083 

.92370 

33698 

.92209 

.40301 

.91519 

.41892 

.90802 

14 

47 

.35484 

.93493 

.37110 

.92859 

.38725 

.92198 

.40328 

.91508 

.41919 

.90790 

13 

48 

.35511 

.93483 

.37137 

.92849 

38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35533 

.93472 

.37164 

.92838 

38778 

.92175 

40381 

.91484 

.41972 

.90766 

11 

50 

.35565 

.93462 

.37191 

.92827 

.38805 

.92164 

.40408 

.91472 

.41998 

.90753 

10 

51 

.a5592 

.93452 

.37218 

.92816 

.38832 

.92152 

.40434 

.91461 

.42024 

.90741 

9 

52 

.35619 

.93441 

.37245 

.92805 

.38859 

.92141 

.40461 

.91449 

.42051 

.90729 

8 

53 

.35647 

.93431 

.37272 

.92794 

.38886 

.92130 

.40488 

.91437 

.42077 

.90717 

7 

54 

.35674 

.93420 

.37299 

.92784 

.38912 

.92119 

.40514 

.91425 

.42104 

.90704 

6 

55 

.35701 

.93410 

.37326 

.92773 

.38939 

.92107 

.40541 

.91414 

.42130 

.90692 

5 

56 

.35728 

.93400 

.37353 

.92762 

.38966 

.92096 

.40567 

.91402 

.42156 

.90680 

4 

57 

.35755 

.93339 

.37380 

.92751 

.38993 

.92085 

.40594 

.91390 

.42183 

.90668 

3 

53 

.35782 

.93379 

.37407 

.92740 

.39020 

.92073 

.40621 

.91378 

.42209 

.90655 

2 

59 

.35810 

.93368 

.37434 

.92729 

.39046 

.92062 

.40647 

.91366 

.42235 

.90643 

1 

60 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91355 

.42262 

.90631 

0 

M 

Cofdn. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Corfu". 

Sine. 

M. 

690 

680 

670 

660 

65° 

TABLE   XVI.       NATURAL    SINES    AND    COSINES. 


291 


350 

360 

aro 

380 

390 

M. 

Sine. 

Costa 

Sine. 

Cosin. 

Sine. 

Costa 

Sine. 

Costa 

Sine 

Costa 

M. 

0 

.42262 

.90631 

.43837 

89379 

.4539 

.8910 

.4694 

.88295 

.4848 

.87462 

60 

] 

.42288 

.90618 

.43863 

.39567 

.45425 

.89087 

,46973 

.8828 

.43506 

.87448 

59 

< 

.42315 

.90506 

.43389 

.39854 

.4545 

.89074 

.4699 

.88267 

.48532 

.87434 

58 

c 

.42341 

.90594 

.43916 

.89341 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

- 

.42367 

.90532 

.43942 

.89823 

.45503 

.89048 

.47050 

.8824! 

.48583 

.87406 

56 

5 

.42394 

90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

45554 

.89021 

.4710 

.88213 

.48634 

S7377 

54 

rt 

42446 

.90545 

.44020 

.89790 

45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

i 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.39752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.4876 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44231 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44335 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

,  22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88808 

.47511 

.87993 

.49040 

.87150 

38 

I  23 

.42867 

90346 

.44437 

.89534 

.45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88763 

.47588 

.87951 

.49116 

.87107 

35 

)  26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

23 

.42999 

.90284 

.44568 

.89519 

.46123 

88728 

.47665 

87909 

.49192 

.87064 

32 

1  29 

.43025 

.90271 

.44594 

.89506 

.46149 

88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

88701 

.47716 

87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89430 

.46201 

88688 

.47741 

87868 

.49268 

.87021 

2V> 

32 

.43104 

.90233 

.44672 

.89467 

.46226 

88674 

47767 

87854 

.49293 

87007 

28 

33 

.43130 

.90221 

.44693 

89454 

.46252 

88661 

47793 

87840 

.49318 

86993 

27 

1  34 

43156  .90208 

.44724 

.89441 

.46278 

88647 

47818 

87826 

.49344 

86978 

26 

1  35 

431821.90196 

.44750 

.89423 

.46304 

88634 

47844 

87812 

.49369 

86964 

25 

1  36 

43209 

.90183 

.44776 

.89415 

.46330 

88620 

47869 

87798 

.49394 

86949 

24 

<  37 

43235 

.90171 

.44802 

.89402 

.46355 

88607 

47895 

87784 

.49419 

86935 

23 

33 

43261 

.90158 

.44828 

89339 

.46381 

88593 

47920 

87770 

49445 

86921 

22 

39 

43287 

.90146 

.44854 

89376 

.46407 

88580 

47946 

87756 

49470 

86906 

21 

40 

43313 

.90133 

.44880 

89363 

46433 

88566 

47971 

87743 

49495 

86892 

20 

41 

43340 

.90120 

.44906 

89350 

.46458 

88553 

47997 

87729 

49521 

86878 

19 

42 

4a366 

.90108 

.44932 

.89337 

.46484 

88539 

48022 

87715 

49546 

86863 

18 

43 

43392 

.90095 

.44958 

89324 

.46510 

88526 

48048 

87701 

49571 

86849 

17 

44 

43418 

.90082 

.44984 

89311 

.46536 

88512 

48073 

87687 

49596 

86834 

16 

45 

43445 

.90070 

45010 

89298 

.46561 

88499 

48099 

87673 

49622 

86820 

15 

46 

43471 

.90057 

45036 

89285 

.46587 

88485 

48124 

87659 

49647 

86805 

14 

47 

43497 

.90045 

45062 

89272 

.46613 

88472 

48150 

87645 

49672 

86791 

13 

48 

43523 

.90032 

45088 

89259 

46639 

88458 

48175 

87631 

49697 

86777 

12 

49 

43549 

.90019 

45114 

89245 

46664 

88445 

48201 

87617 

49723 

86762 

11 

50 

43575 

.90007 

45140 

89232 

46690 

88431 

48226 

87603 

49748 

86748 

Ki 

51 

43602 

.89994 

45166 

89219 

46716 

88417 

48252 

87589 

49773 

86733 

9 

52 

43628 

89981 

45192 

89206 

.46742 

88404 

48277 

87575 

49798 

86719 

8 

53 

43654 

.89968 

45218 

89193 

.46767 

88390 

48303 

87561 

49824 

86704 

7 

54 

43680 

.89956 

45243 

89180 

.46793 

88377 

48328 

87546 

49849 

86690 

6 

65 

43706 

.89943 

45269 

89167 

.46819 

88363 

48354 

87532 

49874 

86675 

5 

56 

43733 

.89930 

45295 

89153 

.46844 

88349 

48379 

87518 

49899 

86661 

4 

57 

43759 

.89918 

45321 

89140 

.46870 

88336 

48405 

87504 

49924 

86646 

3 

68 

43785 

.89905 

45347 

89127 

.46896 

88322 

48430 

87490 

49950 

86632 

2 

59 

43811 

.89892 

45373 

89114 

.46921 

88308 

48456 

87476 

49975 

86617 

I 

60 

43837 

.89879 

45399 

89101 

.46947 

88295 

48481 

87462 

50000 

86603 

0 

sr 

Cofiin. 

Sine. 

Cosin. 

Sine. 

Costa. 

Sine. 

Cosin. 

Sine. 

Coeln. 

Sine. 

M 

640   1 

630 

630 

610 

600 

292 


TABLE   XVI.       NATURAL    SINES   AND    COSINES. 


300 

310 

330 

330 

840 

M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine.  Cosin. 

Sine. 

Cosin. 

Sine. 

Ootdn. 

M. 

0 

50000 

86603 

51504 

.85717 

.62992 

.84805 

.54464 

.83867 

.55919 

.82904 

60 

1 

.60025 

.86588 

51529 

.85702 

.63017 

.84789 

.54488 

.83851 

.55943 

.82887 

59 

2 

.50050 

86573 

51554 

.85687 

.53041 

.84774 

.54513 

.83835 

,55968 

.82871 

58 

3 

.50076 

.86559 

.51679 

.85672 

.53066 

.84759 

.54537 

.83319 

55992 

.82855 

67 

4 

.50101 

.86544 

.51004 

.85067 

.63091 

.84743 

.54561 

.83804 

.56016 

.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.63115 

.84728 

.545.S6 

.83788 

66040 

£2822 

55 

6 

.50151 

.86515 

.61663 

.85627 

.63140 

.84712 

.54610 

.83772 

56064 

.82806 

64 

7 

.50176 

.86501 

.51678 

.86012 

.53164 

.84697 

.54635 

.83756 

,56088 

.82790 

63 

8 

.50201 

.86486 

.51708 

.85597 

.63189 

.84681 

.54659 

.83740 

.66112 

.82773 

52 

9 

.50227 

.86471 

.517X8 

.85682 

53214 

.84666 

.54683 

.83724 

.56136 

.82767 

51 

10 

.50252 

.86457 

.51753 

.85567 

53238 

.84650 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.86551 

53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708 

48 

13 

.50327 

.86413 

.51828 

.85521 

.53312 

.84604 

.54781 

.83660 

.56232 

.82692 

47 

14 

.50352 

.86398 

.51852 

.86606 

.53337 

.84588 

.54805 

.83645 

.66256 

.82675 

46 

15 

.60377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83629 

.66280 

.82659 

46 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.8-1557 

.54854 

.83613 

.56305 

.82643 

44 

17 

.60428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626 

43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610 

42 

19 

.504T8 

.86325 

.51977 

.85431 

.53460 

.84511 

.54927 

.83565 

.66377 

.82593 

41 

20 

.60503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.66401 

.82677 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

.82561 

39 

22 

.50553 

.86281  .52051 

.85385 

.53534 

.84464 

.54999 

.83517 

.56449 

.82544 

38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84418 

.55024 

.83501 

.56473 

.82528 

37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55048 

.83485 

.56497 

.82511 

36 

25 

.60628 

.86237 

.62126 

.85340 

53607 

.84417 

.55072 

.83469 

.56521 

.82495 

35 

26 

.5M54 

.88222 

.52151 

.85325 

53632 

.84402 

.55097 

.83453 

.56545 

.82473 

34 

27 

.50679 

.86207 

52175 

.86310 

.53656 

.84386 

.55121 

.83437 

.56569 

.82462 

S3 

28 

.60704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82429 

31 

30 

.60764 

.86163 

.62250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.66665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.8523-1 

.63779 

.84308 

55242 

.83356 

.66689 

.82380 

28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55266 

.83340 

.66713 

.82363 

37 

34 

.50854 

.86101 

.52349 

.85203 

.53828 

.84277 

.55291 

.83324 

.56736 

.82347 

26 

35 

.60879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.83303 

.56760 

.82330 

26 

36 

.60904 

.86074 

.52399 

.85173 

.53877 

.84245 

55339 

.83292 

.56784 

.82314 

24 

37 

.60929 

.86069 

.52423 

.85157 

.53902 

.84230 

.55363 

.83276 

.56808 

.82297 

23 

33 

.50954 

.86045 

.52448 

.85142 

.53926 

.84214 

.55388 

.83260 

.56832 

.82281 

22 

39 

.60979 

.88030 

.52473 

.85127 

53951 

.84193 

.55412 

.83244 

.56856 

.82264 

21 

40 

.51004 

.86015 

.52498 

.85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.61029 

.86000 

.52522 

.85096 

.54000 

.81167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.61054 

.85985 

.62547 

.85081 

.54024 

.84151 

.55484 

.83195 

.56928 

.82214 

18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509 

.83179 

.56952 

.82198 

17 

44 

.61104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533 

.83163 

.56976 

.82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557 

.83147 

.57000 

.82165 

.16 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.83131 

.67024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.67047 

.82132 

13 

48 

.51204 

.85896 

.52696 

.84989 

.64171 

.84057 

.55630 

.83098 

.67071 

.82116 

12 

49 

.51229 

.85381 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082 

.67095 

.82098 

11 

60 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.65678 

.83066 

.57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050 

.57143 

,82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54269 

.83994 

.55726 

.83034 

.57167 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.65750 

.83017 

.57191 

.82032 

54 

.51354 

.85806 

.52844 

.84897 

.54317 

.83962 

.55775 

.83001 

.57215 

.82015 

t 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.67233 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.54366 

.83930 

.55823 

.82969 

.57262 

.81989 

4 

67 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.65847 

.82953 

67286 

.81965 

3 

58 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.67310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820 

.54440 

.83883 

.65895 

.82920 

.5733-1 

.81932 

1 

60 

.51504 

.85717 

.52992 

.84805 

54464 

.83367 

.55919 

.82904 

.57358 

.81915 

0 

M. 

Cosiii. 

81no. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

M. 

590 

58° 

570 

560 

550 

TABLE   XVI.       NATURAL   SINES   AND    COSINES. 


93° 

360 

3TO 

380 

390 

M. 

Sine, 

Coeln 

Sine. 

Cosin. 

Sine. 

Cosln 

Sine. 

Cosin 

Sine. 

Cceiii 

M. 

0 

.67358 

.81915 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.7771 

60 

.57381 

.81899 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696 

69 

« 

.67405 

.81882 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.7767 

68 

2 

.67429 

.81865 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660 

67 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.63022 

.7764 

56 

5 

.67477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.77623 

66 

| 

.57501 

.81815 

.58920 

.80799 

.60321 

.79768 

.61704 

.78694 

.63068 

.77605 

64 

.57524 

,81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548 

.81782 

58967 

.S0765 

.60367 

.79723 

.61749 

.78658 

.63113 

.7756 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

77560 

61 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.7753 

50 

11 

.57619 

.81731 

.69037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.7751 

49 

12 

.57643 

81714 

.59061 

.80696 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.61864 

.78568 

.63225 

.77476 

47 

14 

.67691 

.81681 

.59108 

.80662 

.60506 

.79618 

.61887 

.78550 

.63248 

.77458 

46 

15 

.67715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532 

.63271 

.77439 

46 

16 

.57738 

.81647 

.59164 

.80627 

.60553 

.79583 

.61932 

.78514 

.63293 

.7742 

44 

17 

.67762 

.81631 

.59178 

.80610 

.60576 

.79565 

.61955 

.78496 

.63316 

.77402 

43 

18 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63338 

.77384 

42 

19 

.67810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.78460 

.63361 

.7736C 

41 

20 

.67833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

.67857 

.81563 

.59272 

.80541 

.60668 

.79494 

.62046 

.78424 

.63406 

.77329 

39 

22 

.67881 

.81546 

.59295 

.80524 

.60691 

.79477 

.62069 

.78405 

.63428 

.77310 

38 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092 

.78387 

.63451 

.77292 

37 

24 

.67928 

.81513 

.59342 

.80489 

.60738 

.79441 

.62115 

.78369 

.63473 

.77273 

36 

26 

67952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255 

36 

26 

67976 

.81479 

.59389 

.80455 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

27 

67999 

.81462 

.59412 

.80438 

.60807 

.79388 

.62183 

.78315 

.63540 

.77218 

33 

28 

68023 

.81445 

.59436 

80420 

.60830 

.79371 

.62206 

.78297 

.63563 

.77199 

32 

29 

68047 

81428 

.59459 

80403 

.60853 

.79353 

.62229 

.78279 

.63585 

.77181 

31 

30 

68070 

81412 

.69482 

80386 

.60876 

.79335 

.62261 

.78261 

.63608 

.77162 

30 

31 

68094 

81395 

.59506 

80368 

.60899 

79318 

.62274 

.78243 

.63630 

.77144 

29 

32 

68118 

81378 

.59529 

80351 

.60922 

.79300 

.62297 

.78225 

.63653 

.77126 

28 

33 

58141 

81361 

.59552 

80334 

.60945 

79282 

.62320 

.78206 

.63675 

.77107 

27 

34 

68165 

81344 

.59576 

80316 

.60968 

79264 

.62342 

.78188 

.63698 

.77088 

26 

35 

681891.81327 

.59599 

80299 

.60991 

79247 

.62365 

.78170 

.63720 

.77070 

25 

36 

582121.81310 

.59622 

80282 

.61015 

79229 

.62388 

.78152 

.63742 

.77061 

24 

37 

582361.81293 

.59646 

80264 

.61038 

79211 

.62411 

78134 

.63765 

.77033 

23 

38 

58260 

.81276 

.59669 

80247 

.61061 

79193 

.62433 

78116 

.63787 

.77014 

22 

39 

68283 

.81259 

.59693 

80230 

.61084 

79176 

.62456 

78098 

.63810 

76996 

21 

40 

68307 

.81242 

.59716 

80212 

.61107 

79158 

62479 

78079 

.63832 

76977 

20 

41 

58330 

.81225 

.59739 

80195 

.61130 

79140 

62502 

78061 

.63854 

76959 

19 

42 

58354 

.81208 

.59763 

80178 

.61153 

79122 

62524 

78043 

.63877 

76940 

18 

43 

58378 

.81191 

.59786 

80160 

.61176 

79105 

62547 

78025 

.63899 

76921 

17 

44 

58401 

.81174 

59809 

80143 

.61199 

79087 

62570 

78007 

.63922 

76903 

16 

45 

58425 

.81157 

59832 

80125 

.61222 

79069 

62592 

77988 

.63944 

76884 

15 

46 

68449 

.81140 

59856 

80108 

61245 

79051 

62615 

77970 

.63966 

76866 

14 

47 

58472 

.81123 

69879 

80091 

.61268 

79033 

62638 

77952 

.63989 

76847 

13 

48 

58496 

.81106 

59902 

80073 

61291 

79016 

62660 

77934 

.64011 

76828 

12 

49 

58519 

.81089 

59926 

80056 

61314 

78998 

62633 

77916 

.64033 

76810 

11 

60 

58543 

.81072 

59949 

80033 

61337 

78980 

62706 

77897 

64056 

76791 

10 

61 

58567 

.81055 

59972 

80021 

61360 

78962 

62728 

77879 

64078 

76772 

9 

52 

58590 

.81038 

59995 

80003 

61383 

78944 

62751 

77861 

64100 

76754 

8 

53 

58614 

.81021 

60019 

799S6 

61406 

78926 

62774 

77843 

64123 

76735 

7 

54 

58637 

.81004 

60042 

79968 

61429 

78908 

62796 

77824 

64145 

76717 

6 

55 

58661 

.80987 

60065 

79951 

61451 

78891 

62819 

77806 

64167 

76698 

6 

56 

58684 

.80970 

60089 

79934 

61474 

78873 

62842 

77788 

64190 

76679 

4 

57 

58708 

.80953 

60112 

79916 

61497 

78855 

62864 

77769 

64212 

76661 

3 

58 

58731 

.80936 

60135 

79899 

61520 

78837 

62887 

77751 

64234 

76642 

2 

59 

58755 

.80919 

60158 

79881 

61543 

78819 

62909 

77733 

64256 

76623 

1 

60 

58779 

.80902 

60182 

79864 

61566 

78801 

62932 

77715 

64279 

.7GC04 

0 

M. 

Cotdn. 

Sine. 

Cosln. 

Sine. 

Coatii. 

Sine. 

Cosin. 

Sine. 

Costn. 

Slue. 

M. 

54o 

530 

5«o 

51° 

50° 

294 


TABLE   XVI.       NATURAL    SINES   AND   COSINES. 


400 

410 

4?o 

430 

440 

BL 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

M. 

o 

.64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

T69466 

.71934 

60 

1 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.63221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.64346 

.76548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 

56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833 

55 

6 

.64412 

.76492 

.65738 

.75356 

.67043 

.74198 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65781 

.75318 

.67086 

.74159 

.63370 

.72976 

.69633 

.71  ?72 

52 

9 

.64479 

.76436 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

.64524 

.76398 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76380 

.65869 

.75241 

.67172 

.74080 

.68455 

.72897 

.69717 

.71691 

48 

13 

.64568 

.76361 

.65891 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

45 

16 

64635 

.76304 

.65956 

.75165 

67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64657 

.76286 

.65978 

.75146 

.67280 

.73983 

.68561 

.72797 

.69321 

.71590 

43 

13 

.64679 

.76267 

.66000 

.75126 

.67301 

.73963 

.68582 

.72777 

.69842 

.71569 

42 

19 

.64701 

.76248 

.66022 

.75107 

.67323 

.73944 

.68603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.76229 

.66044 

.75088 

.67344 

.73924 

.6862-1 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

.75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192 

.66088 

.75050 

.67387 

.73885 

.68666 

.72697 

.69925 

.71488 

33 

23 

.64790 

.76173 

.66109 

.75030 

.67409 

.73865 

.68688 

.72677 

.69946 

.71468 

37 

24 

.64812 

.76154 

.66131 

.75011 

.67430 

.73846 

.68709 

.72657 

.69966 

.71447 

36 

25 

.64834 

.76135 

.66153 

.74992 

.67452 

.73826 

.68730 

.72637 

.69987 

.71427 

35 

26 

.64856 

.76116 

.66175 

.74973 

.67473 

.73806 

.68751 

.72617 

.70008 

.71407 

34 

27 

.64878 

.76097 

.66197 

.74953 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386 

33 

28 

.64901 

.76078 

.66218 

.74934 

.67516 

•73767 

.68793 

.72577 

.70049 

.71366 

32 

29 

.64923 

.76059 

.66240 

.74915 

.67538 

.73747 

.68814 

.72557 

.70070 

.71346 

31 

30 

.64945 

.76041 

.66262 

.74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71326 

30 

31 

.64967 

.76022 

.66284 

.74876 

.67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989 

.76003 

.66306 

.74857 

.67602 

.73688 

.68878 

.72497 

.70132 

.71284 

28 

33 

.65011 

.75984 

.66327 

.74838 

.67623 

.73669 

.68899 

.72477 

.70153 

.71264 

27 

34 

.65033 

.75965 

.66349 

.74818 

.67645 

.73649 

.68920 

.72457 

.70174 

.71243 

26 

35 

.65055 

.75946 

.66371 

.74799 

.67666 

.73629 

.68941 

.72437 

.70195 

.71223 

25 

36 

.65077 

.75927 

.66393 

.74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.71203 

24 

37 

.65100 

.75908 

.66414 

.74760 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182 

23 

38 

.65122 

.75389 

.66436 

.74741 

.67730 

.73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.65144  .75870 

.66458 

.74722 

.67752 

.73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851 

.66480 

.74703 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

.75832 

.66501 

.74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.66523 

.74664 

.67816 

.73491 

.69088 

.72297 

.70339 

.71080 

18 

43 

.65232 

.75794 

.66545 

.74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.71059 

17 

44 

.65254 

75775 

.66566 

.74625 

.67859 

.73452 

.69130 

.72257 

.70381 

.71039 

16 

45 

.65276 

,75756 

.66588 

.74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

.65293 

.75738 

.66610 

.74586 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998 

H 

47 

.65320 

.75719 

.66632 

.74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978 

13 

48 

.65342 

.75700 

.66653 

.74548 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65364 

.75680 

.66675 

.74528 

.67965 

.73353 

.69235 

.72156 

.70484 

.70937 

11 

50 

.65386 

.75661 

.66697 

.74509 

.67987 

.73333 

.69236 

.72136 

.70505 

.70916 

10 

51 

.65408 

.75642 

.66718 

.74489 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896 

9 

52 

.65430 

75623 

.66740 

.74470 

68029 

.73294 

.69298 

.72095 

.70546 

.70875 

8 

53 

.65452 

.75604 

.66762 

.74451 

68051 

.73274 

.69319 

.72075 

.70567 

.70855 

7 

54 

.65474 

.75585 

.66783 

.74431 

68072 

.73254 

.69340 

.72055 

.70587 

.70834 

0 

55 

.65496 

.75566 

.66805 

.74412 

.68093 

.73234 

.69361 

.72035 

,70608 

.70813 

5 

56 

.65518 

.75547 

.66827 

.74392 

.68115 

.73215 

.69332 

.72015 

.70628 

70793 

4 

57 

.65540 

.75528 

.66848 

.74373 

.68136 

.73195 

.69403 

.71995 

.70649 

.70772 

3 

58 

.65562 

.75509 

.66870 

.74353 

.68157 

.73175 

.69424 

.71974 

.70670 

.70752 

2 

59 

.65584 

.75490 

.66891 

.74334 

.68179 

.73155 

.69445 

.71954 

.70690 

.70731 

1 

60 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

.70711 

.70711 

0 

M: 

Costa. 

Sine. 

Cosin. 

SineT 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

M. 

490 

480 

470  i  400 

450 

TABLE  XVII. 


NATURAL  TANGENTS  AND  COTANGENTS. 


296  TABLE   XVII.       NATURAL    TANGENTS    AND    COTANGENTS. 


0° 

10 

»o 

30 

M. 

Ttuig.  |  Cotung. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9756 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711 

58 

3 

.00037 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

,01891 

52.8821 

03638 

27.4899 

05387 

18.5645 

55 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

95416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

05474 

18.2677  52 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503  -  '8.1708 

51 

1C 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533   8.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

05562 

17.9802 

49 

1  12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

05591 

17.8863 

48 

[  13  .00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

05649 

17.7015 

46 

16 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16 

.00465 

214.858 

.0221  1 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

22 

.00640 

156.259 

02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

.02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

30 

.00673 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

SO 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

•15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

IT 

44 

.01280 

78.1263 

.03026 

33.0452 

04774 

20.9460 

.06525 

15.3254 

16  j, 

45 

.01309 

76.3900 

.03056 

32.7303 

.04803 

20.8188 

.06554 

15  2571 

If 

46 

.01338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14  ! 

47 

.01367 

73.1390 

.03114 

32.1131 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.G557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

1! 

5C 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

23.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

M: 

Cotang. 

Tang. 

Cotang. 

Tang. 

Do  tang. 

Tang. 

otang. 

Taug. 

M.  i 

890 

880          870     j 

860 

TABLE   XVIT.       NATURAL    TANGENTS    AND    COTANGENTS.   297 


*o 

50 

60 

TO 

M. 

Ttuig 

Cotang 

Tang. 

Cotang. 

Tang. 

Cotaug. 

Tung. 

Cotanjr. 

M. 

~0 

.06993 

14.31)07 

.08749 

11.4301 

10510 

9.51436 

12278 

8.14436 

60 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.46781 

.12308 

8.12481 

69 

2 

.07051 

14.1821 

.08807 

11.3540 

10569 

9.46141 

.12338 

8.10536 

68 

3 

.07080 

14.1235 

.08837 

11.3163 

10599 

9.43515 

12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

10628 

9.40904 

12397 

8.06674 

56 

6 

.07139 

14.0079 

.08895 

11.2417 

10657 

9.3-3307 

12426 

8.04756 

55 

6 

.07168 

13.9507 

.08925 

11.2048 

10687 

9.35724 

12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

10746 

9.30599 

12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

10775 

9.28058 

12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

1  1  .0594 

10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

10834 

9.23016 

12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9882 

10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

10893 

9.18028 

12662 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

10922 

9.15554 

12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

10952 

9.13093 

12722 

7.86064 

45 

16 

.07461 

13.4039 

.09218 

10.8483 

10981 

9.10646 

.12751 

7.84242 

44 

17 

07490 

13.3515 

.09247 

10.8139 

11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

11040 

9.05789 

12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

11070 

9.03379 

12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

11099 

9.00983 

.12869 

7.77035 

40 

21 

07607 

13.1461 

.09365 

10.6783 

11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

11158 

8.96227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

11217 

8.91520 

.12988 

7.69957 

36 

26 

.07724 

12.9469 

.09482 

10.5462 

11246 

8.89185 

13017 

7.68208 

36 

26 

.07753 

12.8981 

.09511 

10.5136 

11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13078 

7.64732 

33 

28 

.07812 

12.8014 

09570 

10.4491 

11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

11364 

8.79964 

13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

11394 

8.77689 

13165 

7.59575 

30 

31 

.07899 

12.8591 

.09658 

10.3538 

11423 

8.75425 

13195 

7.67872 

29 

32 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7  66176 

28 

33 

.07858 

12.5660 

.09717 

10.2913 

11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

13284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.61132 

26 

36 

.08046 

12.4288 

.09805 

10.1988 

11570 

8.64275 

13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

13372 

7.47806 

23 

38 

.03104 

12.3390 

.09864 

10.1381 

11629 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.09893 

10.1080 

.11659 

8.57718 

13432 

7.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

11688 

8.55555 

13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

11718 

8.53402 

.13491 

7.41240 

19 

421  .08221 

12.1632 

.09981 

10.0187 

11747 

8.51259 

13521 

7.39616 

18 

431  .08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44  |  .08280 

12.0772 

.10040 

9.96007 

11806 

8.47007 

.13580 

7.36389 

16 

45!  .08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

13609 

7.34786 

15 

46  .0,8339 

11.9923 

.10099 

9.90211 

11865 

8.42795 

13639 

7.33190 

14 

47  .08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

48  .08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49  .08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50  i  .08456 

11.8262 

.10216 

9.78817 

11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

9.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

12042 

8.30406 

13817 

7.23754 

8 

53  i  .08544 

11.7045 

.10305 

9.70441 

12072 

8.28376 

.13846 

7.22204 

7 

54  i  .08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

12131 

8.24345 

.13906 

7.19125 

5 

56 

08632 

11.5853 

.10393 

9.62205 

12160 

8.22344 

.13935 

7.17594 

4 

57 

08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

'.16071 

3 

58 

Ort690 

11.5072 

.10452 

9.56791 

12219 

8.18370 

13995 

*  14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

6U  .08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

M.  Getting.  Tbug. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 

!     650 

840 

830 

8»o 

£98  TABLE   XVII.       NATUKAL    TANGENTS    AND    COTANGENTS. 


8° 

9° 

100 

11° 

M 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

^li 

~0 

14054 

14084 

7.11537 
7  10038 

15838 
15868 

6.31375 
6.30189 

17633 
17663 

5.67128 
5.66165 

.19438 
.19468 

5.14455 
5.13658 

60 
59 

2 

.14113 
14143 

7.08546 
7.07059 

15898 
15928 

6.29007 

6.27829 

17693 
17723 

5.65205 
5.64248 

.19498 
.19529 

5.12862 
5.12069 

bb 
57 

14173 

7  05579 

15958 

6.26655 

17753 

5.63295 

.19559 

5.  1  1279 

56 

g 

14202 

7.04105 

15988 

6.25486 

17783 

5.62344 

.19589 

5.  10490 

bb 

v  14232 

7.02637 

16017 

6.24321 

17813 

5.61397 

.19619 

5.09704 

64  | 

7 
3 
9 
0 

13 
14 
15 

.14262 
.14291 
.14321 
.14351 
.14381 
.14410 
.14440 
.14470 
.14499 

7.01174 
6.99718 
0.  98268 
6.96823 
6.95385 
6.93952 
6.92525 
6.91104 
6.89688 

.16047 
16077 
.16107 
.16137 
16167 
.16196 
.16226 
.16256 
.16286 

6.23160 
6.22003 
6.20851 
6.19703 
6.18559 
6.17419 
6.16283 
6.15151 
6.14023 

.17843 
.17873 
.17903 
.17933 
.17963 
.17993 
.18023 
.18053 
.18083 

5.60452 
5.59511 
5.58573 
5.57638 
5.56706 
5.55777 
5.54851 
5.53927 
5.53007 

.19649 
.19680 
.19710 
.19740 
.19770 
19801 
.19831 
.19861 
.19891 

5.08921 
5.08139 
5.07360 
5.06584 
5.05809 
5.05037 
5.04267 
5.03499 
6.02734 

b3 
52 
51 
50 
49 
48 
47 
46 
45 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

.14529 
.14559 
.14588 
.14618 
.14648 
.14678 
.14707 
.14737 
.14767 
.14796 
.14326 
.14856 
.14886 
.14916 
.14945 

6.88278 
6.86874 
6.85475 
6.84082 
6.82694 
6.81312 
6.79936 
6.78564 
6.77199 
6.75838 
6.74483 
6.73133 
6.71789 
6.70450 
6.69116 

.16316 
.16346 
16376 
.16405 
.16435 
.16465 
16495 
.16525 
.16555 
.16585 
.16615 
.16645 
.16674 
.16704 
.16734 

6.12899 
6.11779 
6.10664 
6.09552 
6.0S444 
6.07340 
6.06240 
6.05143 
6.04051 
6.02962 
6.01878 
6.00797 
5.99720 
5.98646 
5.97576 

.18113 
.18143 
.18173 
.18203 
.18233 
.18263 
.18293 
.18323 
.18353 
.18384 
.18414 
.18444 
.18474 
.18504 
.18534 

6.52090 
5.51176 
5.50264 
5.49356 
5.48451 
5.47548 
5.46648 
6.45751 
5.44857 
5.43966 
5.43077 
5.42192 
6.41309 
6.40429 
6.39552 

.19921 
.19952 
.19982 
.20012 
.20042 
.20073 
.20103 
.20133 
.20164 
.20194 
.20224 
.20254 
.20285 
.20315 
.20345 

6.01971 
5.01210 
6.00451 
4.99695 
4.98940 
4.98188 
4.97438 
4.96690 
4.95945 
4.95201 
4.94460 
4.93721 
4.92984 
4.92249 
4.91516 

44 
43 
42 
41 
40 
39  ! 
38 
37  i 
36 
35 
34  I 
33 
32 
31 
30  - 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 

.14976 
.15005 
.16034 
.15064 
.15094 
.16124 
.15153 
.15183 
.15213 
.15243 
.15272 
.15302 
.15332 
.15362 
.15391 

6.67787 
6.66463 
6.65144 
6.63831 
6.62523 
6.61219 
6.59921 
6.58627 
6.57339 
6.56055 
6.54777 
6.53503 
6.52234 
6.50970 
6.49710 

.16764 
.16794 
.16824 
.16854 
.16884 
16914 
.16944 
.16974 
.17004 
.17033 
.17063 
.17093 
.17123 
.17153 
.17183 

6.96510 
5.95448 
5.94390 
5.93335 
5.92283 
5.91236 
6.90191 
5.89151 
6.88114 
5.87080 
5.86051 
5.85024 
6.84001 
5.82982 
5.81966 

.18564 
.18594 
.18624 
.18654 
.18684 
.18714 
.18745 
.18775 
.18805 
.18835 
.18865 
.18895 
.18925 
.18955 
.18986 

6.38677 
5.37805 
5.36936 
5.36070 
5.35206 
5.34345 
5.33487 
5.32631 
6.31778 
5.30928 
6.30080 
5.29235 
5.28393 
5.27553 
6.26715 

.20376 
.20406 
.20436 
.20466 
.20497 
.20527 
.20557 
.20588 
.20618 
.20648 
.20679 
.20709 
.20739 
.20770 
.20800 

4.90785 
4.90056 
4.89330 
4.88605 
4.87882 
4.87162 
4.86444 
4.85727 
4.85013 
4.84300 
4.83590 
4.82882 
4.82176 
4.81471 
4.80769 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 

46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
66 
57 
58 
59 
60 

.15421 
.15451 
15481 
.15511 
.15540 
.15570 
.15600 
.15630 
.15660 
.15689 
.16719 
.16749 
.15779 
.15809 
.15838 

6.48456 
6.47206 
6.45961 
6.44720 
6.43484 
6.42253 
6.41026 
6.39804 
6.38587 
6.37374 
6.36165 
6.3496 
6.3376 
6.32566 
6.31375 

.17213 
.17243 
.17273 
.17303 
.17333 
.17363 
.17393 
.17423 
17453 
.17483 
.17513 
.17543 
.17573 
.17603 
.17633 

5.80953 
5.79944 
5.78938 
5.77936 
5.76937 
5.75941 
5.74949 
5.73960 
5.72974 
5.71992 
6.71013 
5.70037 
5.69064 
5.68094 
5.67128 

.19016 
19046 
.19076 
.19106 
.19136 
.19166 
.19197 
.19227 
.19257 
.19287 
.19317 
.19347 
.19378 
.19408 
.19438 

5.25880 
5.25048 
5.24218 
6.23391 
5.22566 
6.21744 
6.20925 
5.20107 
6.19293 
5.18480 
6.17671 
6.16863 
5.16058 
6.15256 
6.14455 

.20830 
.20861 
.20891 
.20921 
.20952 
.20982 
.21013 
.21043 
.21073 
.21104 
.21134 
.21164 
.21195 
.21225 
.21256 

4.80068 
4.79370 
4.78673 
4.77978 
4.77286 
4.76595 
4.75908 
4.76219 
4.74534 
4.73851 
4.73170 
4.72490 
4.71813 
4.71137 
4.70463 

14 
13 
12 
11 
10 
9 
8 
7 
6 
5 

3 

2 

0 

K 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

T*ng. 

Co  twig 

Tung. 

ML 

810 

800 

TOO 

w 

TABLE   XVII.       NATURAL   TANGENTS   AND    COTANGENTS.   299 


ia° 

13° 

140 

150 

If 

Tang. 

Cotang 

Tung. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

M. 

0 

21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.00582 

.26826 

3.72771 

69 

2 

.21316 

4.69121 

.23148 

4.32001 

.24395 

4.00086 

.26857 

3.72338 

68 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

66 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

64 

7|  -21469 

4.65797 

.23301 

4.29159 

25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.21529 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

5] 

10 

21560 

4  63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

27169 

3.68061 

48 

13 

21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

21712 

4.60572 

23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

46 

16 

21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.592H3 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

27357 

3.65538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64706 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20842 

25614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

27545 

3.63048 

36 

25 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

36 

26 

.22047 

4.53568 

•23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

25862 

3.86671 

.27732 

3.60688 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59776 

28 

33 

22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

22292 

4.43600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

35 

22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

26 

36 

22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.64573 

16 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

43 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.3S969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

62 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4  04586 

26577 

3.76268 

.28454 

3.51441 

7 

64 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

28486 

3.51053 

65 

.22934 

4.36040 

24778 

4.03578 

.26639 

3.75388 

28517 

3.50666 

66 

.22964 

4.35459 

24809 

4.03076 

.26670 

3.74950 

28549 

3.50279 

57 

.22995 

4.34879 

24840 

4.02574 

26701 

3.74512 

28580 

3.49894 

68 

23026 

4.34300 

24871 

4.02074 

.26733 

3.74075 

28612 

3.49509 

69 

.23056 

4.33723 

24902 

4.01576 

26764 

3.78640 

28643 

3.49125 

60 

.23087 

4.33148 

24933 

4.01078 

26795 

3.73205 

28675 

3.48741 

0 

M: 

Cotoug. 

Tang. 

otang. 

Tang. 

ctang. 

Tang. 

otang. 

Taiig. 

M. 

770 

760 

750 

740 

300  TABLE    XVII.       NATURAL     TANGENTS    AND    COTANGENTS. 


100 

17° 

18° 

190 

1 

M. 

Taiig. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotaug. 

M. 

~o 

.28675 

3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

30637 

3.26406 

32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.2606? 

.32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

32621 

3.06554 

.34563 

2.89327 

56 

5 

,28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

65 

6 

,28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

,32717 

3.05649 

34661 

2.88511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

g 

.28958 

?.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2.87700 

50 

U 

.29021 

3.44576 

.30923 

3.23381 

32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

32878 

3.04152 

34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16 

29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17  !  29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

18  .29242 

3.41973 

.31147 

3.21063 

.33072 

3.B2372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40  : 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

26 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

318127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33396 

2.99447 

.35346 

2.82914 

32  i 

29 

.29590 

3.37965 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16617 

.33524 

2.98292 

.36477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33657 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

..31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

26 

36 

.2981  1 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.  14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11776 

34010 

2.94028 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11  ! 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.0929S 

.34270 

2.91799 

.36232 

2.75996 

6 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

M: 

Cotaug. 

Tang. 

Cotang. 

Tang 

Cotang. 

Tang. 

Cotaug. 

Ttuig. 

M 

T3o 

T«O 

yio 

TOO 

TABLE  XVII.   NATURAL  TANGENTS  AND  COTANGENTS.  301 


|     200 

210 

230 

230 

M.  Tang. 

Cotang. 

Tang. 

Cotang 

Tang 

Cotang 

Tang. 

Cotang 

M. 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

6( 

1 

.36430 

2.74499 

.38420 

2.6028 

.4043 

2.47302 

.42482 

2.35395 

69 

2 

.36463 

2.74251 

.38453 

2.6005 

.40470 

2.47095 

.42516 

2.35205 

5* 

3 

.36496 

2.74004 

.38487 

2.5983 

.40504 

2.46838 

.42551 

2.35016 

57 

4 

.36529 

2.73756 

.38520 

2.5960 

.40538 

2.46682 

.42585 

2.34825 

56 

6 

.36562 

2.73509 

.38553 

2.5938 

.40572 

2.46476 

.42619 

2.34636 

55 

6 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

5i 

8 

.36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.5826 

.40741 

2.45451 

.42791 

2  33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33506 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46 

15 

.36892 

2.71062 

.33888 

2.57150 

.4091  1 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36991 

2.70335 

.38988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

36 

26 

.37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2.67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.6770U 

39357 

2.  54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

^9425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66989 

^58 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

3*49* 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.3952H 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

37554 

2.66281 

.39559 

252786 

.41592 

2.40432 

.43654 

2.29073 

26 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

252357 

.41660 

2.40038 

43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

22 

|  39 

.37687 

2.65342 

.39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.5J715 

41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

41831 

2.39058 

.43897 

2.27806 

18 

43  .37820 

2.64410 

.39829 

2.51076 

41865 

2.38863 

43932 

2.27626 

17 

14  ;  37353 

2.64177 

.39862 

2.50864 

41899 

2.38668 

43966 

2.27447 

16 

15  .37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

44001 

2.27267 

15 

16   37920 

2.63714 

.39930 

2.50440 

41968 

2.38279 

44036 

2.27088 

4 

17 

37853 

2.63433 

39963 

2.50229 

.42002 

2.38084 

44071 

2.26909 

3 

43 

.37986 

2.63252 

39997 

250018 

.42036 

2.37891 

44105 

2.26730 

2 

49 

.38020 

2.63021 

40031 

2.49807 

42070 

2.37697 

44140 

2.26552 

50 

.38053 

2.62791 

40065 

2.49597 

42105 

2.37504 

44175 

2.26374 

0 

51  .380,86 

262561 

40098 

2.49386 

42139 

2.37311 

44210 

2.26196 

9 

52  i  38120 

262332 

40132 

2.49177 

42173 

2.371  18 

44244 

2.26018 

8 

53  |  38153 

2.62103 

.40166 

2.48967 

42207 

2.36925 

44279 

2.25840 

7 

54  .38186 

2.61874 

.40200 

2.48758 

42242 

2.36733 

44314 

2.25663 

6 

55  .3S220 

2.61646 

4023-1 

2.48549 

42276 

2.36541 

44349 

2.25486 

6 

56 

.3S253 

2.61418 

40267 

2.48340 

42310 

2.36349 

44384 

2.25309 

4 

57 

.38286 

261190 

40301 

2.48132 

42345 

2.36158 

44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

42379 

2.35967 

44453 

2.24956 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

42413 

2.35776 

44488 

2.24780 

1 

60  .38386 

2.60509 

.40103 

2.47509 

42447 

2.35585 

44523 

2.24604 

_0 

M.  Co  tang. 

Tang. 

Dotang.  1  Tang. 

otang. 

Tang. 

otang. 

Tang. 

!     690 

680 

670 

660 

H 

302  TABLE   XVII.       NATURAL    TANGENTS    AND    COTANGENTS. 


~34^  

350 

360 

3TO 

M. 

Tang. 

Cotang. 

Tang. 

Cotaug. 

Tang. 

Cotang. 

Toiig 

Cotang. 

M. 

IF 

.44523 

2.24604 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

60 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

1.96120 

59 

2 

.44593 

2.24252 

.467(12 

2.14125 

.48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698 

56 

5 

.44697 

2.23727 

46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

Q 

.44732 

2.23553 

.46343 

2.13477 

48989 

2.04125 

.51173 

1.95417 

54 

7 

!  44767 

2^23378 

.46879 

2.13316 

.49026 

2.03975 

51209 

1.95277 

53 

Q 

.44802 

2  23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

g 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

222857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

60 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

51356 

1.94718 

49 

12 

44942 

222510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

45012 

2  22164 

.47128 

2.12190 

.49278 

2.02929 

.61467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.61540 

1.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.61614 

1.93746 

42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.61651 

1.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.61688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.0189J 

.51724 

1.93332 

39 

22 

45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.61761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.930D7 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

36 

26 

.45432 

2.20108 

.47555 

2.10284 

49713 

2.01155 

.51909 

1  92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

I.  92508 

33 

28 

45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.61983 

1.92371 

32 

29 

.46538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.62057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52054 

1.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.62279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52353 

1.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.60222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

1.98684 

.52538 

1.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.60368 

1.98540 

.-52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

1.98110 

.52687 

1  .89801 

13 

48 

.46206 

2.16420 

48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2  16255 

48378 

2.06706 

.50550 

1.97823 

.52761 

1.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.89266 

9 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

6 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

57 

.46525 

2  14940 

48665 

2.05485 

.50843 

1.96685 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

.50879 

1.96544 

.53096 

1.88337 

2 

59 

.46595 

2  14614 

48737 

2.05182 

.50916 

1.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

.53171 

1.88073 

0 

M 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cofcang. 

Tang. 

Cotang. 

Tung. 

M 

600 

640 

630 

oao 



TABLE    XVII.       NATURAL    TANGENTS    AND    COTANGENTS.  303 


HI 

390 

300 

310 

M. 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

c 

.6317 

1.8807 

.5543 

T.8040 

.57735 

1.7320 

.6008 

1.6642 

6U 

] 

.63208 

1.8794 

.55469 

1.8028 

.5777 

1.7308 

.6012 

1.6631 

5 

2 

.63246 

1.8780 

.55507 

1.8015 

.5781 

1.7297 

.6016 

1.6620 

5 

3 

.53283 

1.8767 

.55545 

1.80034 

.5785 

1.7285 

.6020 

1.66099 

5! 

4 

.63320 

1.8754 

55583 

1.7991 

.5789 

1.7274 

6024 

l.6599i 

5, 

5 

.63358 

1.8741 

.55621 

1.79788 

.5792 

1.72620 

60284 

1.6588 

5 

6 

.53395 

1.87283 

.55659 

1.7966 

.57968 

1.7250 

.60324 

1.65772 

& 

7 

.63432 

1.8715 

.55697 

1.7954 

.5800 

1.7239 

60364 

1.65663 

5J  ! 

8 

.63470 

1.8702 

.55736 

1.7941 

.58046 

1.7227 

.60403 

1.655& 

5! 

9 

.63507 

1.8689 

.55774 

1.79296 

.58085 

1.7216 

.60443 

1.65445 

51 

10 

63545 

1.86760 

.55812 

1.79174 

.58124 

1.7204 

.60433 

1.65337 

5( 

11 

53582 

1.86630 

.55850 

1.7905 

.58162 

1.71932 

60522 

1.65228 

41 

12  .53320 

1.86499 

.55888 

1.78929 

.58201 

1.7181 

.60562 

1.65120 

41 

13  63657 

1.86369 

.55926 

1.78807 

.58241 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55964 

1.78685 

.58279 

1.71588 

.60642 

1.6490J 

41 

16 

.53732 

1.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.63769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53807 

1.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.64579 

4' 

18 

.53844 

1.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

4' 

19 
20 

.53882 
.53920 

1.85591 
1.85462 

.56156 
.56194 

1.78077 
1.77955 

58474 
.58513 

1.71015 
1.70901 

.60841 
.60881 

1.64363 
1.64256 

41 
40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

22 

.63995 

1.85204 

.56270 

1.77713 

.58591 

1.70673 

.60960 

1.64041 

38 

23 

.64032 

1.85075 

.56309 

1.77592 

.58631 

1.70560 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

.66347 

1.77471 

.58670 

1.70446 

.61040 

1.63826 

36 

26 

.54107 

1.84818 

.66385 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

36 

26 

.64145 

1.84689 

.56424 

1.77230 

.58748 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

1.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1  63505 

33 

28 

.54220 

1.84433 

.66501 

1.76990 

58826 

1.69992 

.61200 

1.63398 

32 

29 

.64258 

1.84305 

56539 

1.76869 

58865 

1.69879 

.61240 

1.63292 

31 

30 

.64296 

1.84177 

66677 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

54333 

1.84049 

56616 

1.76629 

58944 

1.69653 

61320 

1.63079 

29 

32 

.64371 

1.83922 

56654 

1.76510 

58983 

1.69541 

61360 

1.62972 

28 

33 

.64409 

1.83794 

66693 

1.76390 

59022 

1.69428 

61400 

1.62866 

27 

34 
36 

.54446 

.64484 

1.83667 
1.83540 

56731 
56769 

1.76271 
1.76151 

59061 
59101 

1.69316 
1.69203 

61440 
61480 

1.62760 
1.62654 

26 

25 

36 

.64522 

1.83413 

56808 

1.76032 

69140 

1.69091 

61520 

1.62548 

24 

(7 

.64560 

1.83286 

56846 

1.75913 

59179 

1.68979 

61561 

1.62442 

23 

38 

.64597 

1.83159 

66885 

1.75794 

59218 

1.68866 

61601 

1.62336 

22 

39 

.64635 

1.83033 

56923 

1.75675 

59258 

1.68754 

61641 

1.62230 

21 

0 

.64673 

1.82906 

56962 

1.75556 

59297 

1.68643 

61681 

1.62125 

20 

\   I 

.64711 

1.82780 

57000 

1.75437 

59336 

1.68531 

61721 

1.62019 

19 

\  42 

.64748 

1.82654 

57039 

1.75319 

59376 

1.68419 

61761 

1.61914 

8 

1  3 

.64786 

1.82528 

57078 

1.75200 

59415 

1.68308 

61801 

1.61808 

7 

I  44 

.54824 

1.82402 

57116 

1.75082 

59454 

1.68196 

61842 

1.61703 

6 

46 

.64862 

1.82276 

57155 

1.74964 

59494 

1.68085 

61882 

1.61598 

5 

46 

.54900 

1.82150 

57193 

1.74846 

59533 

1.67974 

61922 

1.61493 

4 

7 

.54938 

1.82025 

57232 

1.74728 

59573 

1.67863 

61962 

1.61388 

3 

8 

.64975 

1.81899 

57271 

1.74610 

59612 

1.67752 

62003 

1.61283 

2 

1 

.65013 

1.81774 

57309 

1.74492 

59651 

1.67641 

62043 

1.61179 

1 

5C 

.65051 

1.81649 

57348 

1.74375 

59691 

1.67530 

62083 

1.61074 

0 

51 

.65089 

1.81524 

57386 

1.74257 

59730 

1.67419 

62124 

1.60970 

9 

62 

.65127 

1.81399 

57425 

1.74140 

59770 

1.67309 

62164 

.60865 

8 

>3 

.55165 

81274 

57464 

1.74022 

69809 

1.67198 

62204 

1.60761 

7 

A 

.55203 

1.81150 

57503 

1.73905 

59849 

1.67088 

62245 

1.60657 

6 

66 

.55241 

1.81025 

57541 

1.73788 

59888 

1.66978 

62285 

.60553 

6 

66 

.6527$ 

.80901 

57580 

1  73671 

59928 

1.66867 

62325 

.60449 

4 

57 

.55317 

.80777 

57619 

1.73555 

59967 

.66757 

62366 

.60345 

3 

68 

.55355 

.80653 

57657 

1  73438 

60007 

.66647 

62406 

.60241 

2 

69 

.65393 

.80529 

57696 

1  73321 

600-16 

.66538 

62446 

.60137 

1 

CO 

.55431 

.80405 

57735 

1  73205 

60086 

.66428 

62487 

.60033 

0 

M.  Cotang. 

Taug. 

otang. 

Tang. 

otang. 

Tang. 

otang. 

Tang. 

610 

603 

590 

580 

304  TABLE   XVII.       NATURAL    TANGENTS    AND    COTANGENTS. 


8«o 

330 

M 

Tang. 

Cotaug. 

Tang. 

Cotang. 

long. 

Cotang. 

Taug. 

Cotang. 

M. 

~o 

62487 

1.60033 

64941 

1.53986 

67451 

1.48256 

.70021 

1.42815 

60  ' 

] 

.62527 

1.59930 

64982 

1.53888 

.67493 

1.48163 

.70064 

1  .42726 

59 

2 

62568 

1  59826 

65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

63  ! 

3 

62608 

1.59723 

65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

67 

4 

.62649 

1.59620 

65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56  ' 

5 

.62689 

1.59517 

65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55  , 

6 

.62730 

1.59414 

65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54  ! 

7 

.62770 

1.59311 

65231 

1.53302 

.67748 

1.47607 

70325 

1.42198 

53 

g 

.62811 

1.59208 

65272 

1.53206 

.67790 

1.47514 

.70368 

1.42110 

52 

Q 

62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

|l 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

70499 

1.41847 

49 

12 

.62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

63055 

1.58593 

.65521 

1.52622 

.68045 

1.46962 

.70629 

1.41584 

46 

15 

.63095 

1.58490 

65563 

1.52525 

.68088 

1.46870 

.70673 

1.41497 

45 

16 

63136 

1.58338 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17 

.63177 

1.58286 

.65646 

1.52332 

.68173 

1.46686 

.70760 

1.41322 

43 

18 

63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.61946 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57778 

.65854 

1.51860 

.68386 

1.46229 

.70979 

1.40887 

38 

23 

.63421 

1.57676 

.65896 

1.51754 

.68429 

1.46137 

71023 

1.40800 

37 

24 

.63462 

1.67575 

.65938 

1.51658 

.68471 

1.46046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.65980 

1.61562 

.68514 

1.45955 

.71110 

1.40627 

36  I 

26 

63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

71154 

1.40540 

34 

27 

.63584 

1.57271 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33  ! 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

29 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

31 

80 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27  ! 

84 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

26 

36 

.63953 

1.56366 

.66440 

1.50512 

.68985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.56065 

.66566 

1.50228 

.69114 

1.44688 

71725 

1.39421 

21  j 

40 

.64117 

1.55966 

.66608 

1  50133 

.69157 

1.44598 

71769 

1.39336 

20  ! 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1  49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1  49755 

.69329 

1.44239 

.71946 

1.38994 

16  ! 

45 

.64322 

1.55467 

.66818 

1.49661 

69372 

1.44149 

.71990 

1.38909 

16 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14  I 

47 

64404 

1  55269 

66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12 

49 

.64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

72211 

1.38484 

10 

51 

.64569 

1.54873 

,67071 

1.49097 

.69631 

1.43614 

72255 

1.38399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

72299 

1.38314 

8 

63 

.64652 

1.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

1.54576 

.67197 

1.48816 

.69761 

1  .43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

5 

56 

.64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

57 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

58 

64858 

1  54183 

67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

69 

.64899 

1.54085 

.67409 

i.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

60 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

0 

M 

Ootaug 

Taug. 

Cotang 

Tang 

Cotauff 

Tang. 

Cotang 

Tang. 

M. 

57° 

560 

55° 

6*0 

i 

TABLE   XVII.      NATURAL    TANGENTS    AND    COTANGENTS,  3Q5 


390 

1  —- 

Tang 

Cotang 

Tang 

Cotan 

Tang 

Cotan 

Tang. 

Cotang. 

M. 

1 

.72654 
.7269 

1.37638 
1.37554 

.7535 
.7540 

1.32704 
1.32624 

.7812 
.7817 

1.27994 
1.2791 

.80978 
.81027 

1.23490 
1  2341 

60 
59 

i 

.7274 
.72788 
.72832 

.72877 

1.3747 
1.37386 
1.37302 
1.3721 

.7544 
.75492 
.75538 

.76584 

1.3254 
1.32464 
1.32384 
1.32304 

.78222 
.7826 
.7831 
.7836 

1.2784 
1.27764 
1.27688 
1.2761 

.81075 
.81123 
.81171 
.81220 

1.2334 
1.2327 
1.23196 
1  23123 

58 
57 
66 
55 

I  6 

* 
9 

.7292 
.72966 
.73010 
.73055 

1.37134 
1.37050 
1.3696 
1.36883 

.7562 
.76676 
.7572 
.75767 

1.3222 
1.3214 
1.32064 
131984 

.78410 
.7846 

.78504 
.7855 

1.27535 
1.27458 
1.27382 
1.27306 

.81268 
.81316 
.81364 
.81413 

1.2305 
1.2297 
1.22904 
1  2283 

64 
53 
52 
51 

10 
11 
12 
13 

.73100 
.73144 
.73189 
.73234 

1.36800 
1.3671 
1.3663 
1.3654 

.75812 
.75858 
.75904 
.75950 

1.31904 
1.31825 
1.3174 
1.31666 

.78598 
.78645 
.78692 
.78739 

1.27230 
1.27163 

1.2707 
1.2700 

.81461 
.81510 
.81558 
.81606 

1.22758 
1.22685 
1.22612 
1  22539 

60 
49 
48 
47 

14 
16 

.73278 
.73323 

1.36466 
1.36383 

.75996 
.76042 

1.31586 
1.31507 

.78786 
.78834 

1.26925 
1.2684 

.8165 
.81703 

1.22467 
1.22394 

46 
46 

]  16 
17 
18 
19 
20 
21 
22 
23 
24 
26 
26 
27 
28 
29 
30 

.73368 
.73413 
.73457 
.73502 
.73547 
.73592 
.73637 
.73681 
.73726 
.73771 
.73816 
.73861 
.73906 
.73951 
.73996 

1.36300 
1.3621~ 
1.36134 
1.3605 
1.35968 
1.35886 
1.35802 
1.35719 
1.35637 
1.35554 
1.35472 
1.35389 
1.35307 
1.35224 
1.35142 

.76088 
.76134 
.76180 
.76226 
.76272 
.76318 
.76364 
.76410 
.76456 
.76502 
.76548 
.76594 
.76640 
.76686 
,76733 

1.31427 
1.3134 
1.3126 
1.3119 
1.31110 
1.3103 
1.30952 
1.30873 
1.30795 
1.30716 
1.30637 
1.30558 
1.30480 
1.30401 
1.30323 

.78881 
.78928 
.78976 
.79022 
.79070 
.79117 
.79164 
.79212 
.79269 
.79306 
.79364 
.79401 
.79449 
.79496 
.79644 

1.2677 
1.26693 
1.2662 
1.2654 
1.2647 
1.2639 
1.2631 
1.26244 
1.2616 
1.2609 
1.26018 
1.25943 
1.25867 
1.25792 
1.25717 

.81752 
.81800 
.81849 
.81898 
.81946 
.81995 
.82044 
.82092 
.82141 
.82190 
.82238 
.82287 
.62336 
.82385 
.82434 

1.2232 
1.22249 
1.22176 
1.22104 
1.22031 
1.21959 
1.21886 
1.21814 
1.21742 
1.21670 
1.21598 
1.21626 
1.21454 
1.21382 
1.21310 

44 
43 
4 

4i 
31 

8! 

31 
34 

K 

11 
30 

31 
32 
33 
34 
36 
36 
37 
38 
39 
40 

42 
43 
44 
jl  46 

74041 
.74086 
.74131 
74176 
.74221 
74267 
74312 
74357 
74402 
74447 
74492 
74533 
74583 
74628 
74674 

1.35060 
1.34978 
1.34896 
1.34814 
1.34732 
1.34650 
1.34568 
1.34487 
1.34405 
1.34323 
1.34242 
1.34160 
1.34079 
1.33998 
1.33916 

.76779 
76825 
76871 
76918 
76964 
77010 
77057 
77103 
77149 
77196 
77242 
77289 
77335 
77382 
77428 

1.30244 
1.30166 
1.30087 
1.30009 
1.29931 
1.29853 
1.29775 
1.29696 
1.29613 
1.29541 
1.29463 
1.29385 
1.29307 
1.29229 
1.29152 

.79591 

.79639 
.79686 
.79734 
.79781 
79829 
79877 
79924 
79972 
80020 
80067 
80115 
80163 
80211 
80258 

1.25642 
1.25667 
1.25492 
1.26417 
1.25343 
1.26268 
1.25193 
1.25118 
1.25044 
1.24969 
1.24895 
1.24820 
1.24746 
1.24672 
1.24597 

.82483 
.82531 
.82580 
82629 

1.21238 
1.21166 
1.21094 
1.21023 
1.20961 
1.20879 
1.20808 
1.20736 
1.20665 
1.20593 
1.20522 
1.20451 
1.20379 
1.20308 
1.20237 

g 
27 
26 
26 
24 
23 
22 
21 
20 
19 
8 
17 
6 
6 

.82678 
82727 
82776 
82825 
82874 
82923 
82972 
83022 
83071 
83120 
83169 

46 
47 
48 
49 
60 
61 

1  KQ 

74719 
.74764 
.74810 
.74855 
.74900 
.74946 

1.33835 
1.33754 
1.33673 
1.33592 
1.33511 
1.33430 

77475 
77521 
77568 
77615 
77661 
77708 

1.29074 

1.28997 
1.28919 
1.28842 
1.28764 
1.28687 

80306 
80354 
80402 
80450 
80498 
80546 

1.24523 
1.24449 
1.24375 
1.24301 
1.24227 
1.24163 

83218 
83268 
83317 
83366 
83416 
83465 

1.20166 
1.20095 
1.20024 
1.19953 
1.19882 
1.19811 

4 

3 
2 
1 
0 
9 

1  1  eo 

.74991 

1.33349 

77754 

.28610 

80594 

1.24079 

83614 

1.19740 

64 
65 

.75037 
.75082 
.75128 

1.33263 
1.33187 
.33107 

77801 
77848 
77895 

.28533 
.28456 
.28379 

80642 
80690 
80738 

1.24006 
1.23931 
1.23858 

83564 
83613 
83662 

1.19669 
1.19599 
1  19528 

7 
6 
5 

66 
67 
68 
69 
60 

"53T  7 

.75173 
.75219 
.76264 
.75310 
.76355 
.   — 

.33026 
.32946 
.32865 
.32785 
.32704 

77941 
77988 
78035 

78082 
78129 

.28302 
.28225 
.28148 
.28071 
.27994 

80786 
80834 
80882 
80930 
80978 

.23784 
.23710 
.23637 
.23563 
.23490 

83712 
83761 
83811 
83860 
83910 

1.19457 
1.19387 
1.19316 
1.19246 
1.19176 

4 
3 

2 

M  .  ( 

bteng. 

Tang. 

tang. 

Tang. 

tang. 

Tang. 

tang. 

Tan*. 

530    | 

510   - 

500 

306  TABLE   XVII.       NATURAL    TANGENTS    AND    COTANGENTS. 


1  — 

P 

41 

o 

42 

0 

43 

M. 

Tang. 

otang. 

ang. 

otang. 

ang. 

otang. 

ang. 

otang.  M. 

I 

\ 

83910 
83960 
84009 
84059 
84108 
84158 

19175 
19105 
.19035 
.18964 
.18894 
.18824 

86929 
6980 
7031 
7082 
87133 
87184 

15037 
.14969 
.14902 
.14834 
.14767 
.14699 

0040 
0093 
0146 
0199 
0251 
90304 

11061 
10996 
.10931 
.10867 
.10802 
.10737 

3252 
3306 
3360 
3415 
93469 
93524 

.07237  60 
.07174  59 
.07112  58 
.07049  57 
.06987  56 
.06925  55 

I 

• 

i 

14 
15 

84208 
84258 
84307 
84357 
84407 
84457 
84507 
84556 
84606 
84656 

.18754 
.18684 
.18614 
.18544 
.18474 
.18404 
.18334 
.18264 
.18194 
.18125 

87236 
87287 
87338 
87389 
87441 
87492 
87543 
87595 
87646 
87698 

.14632 
.14565 
.14498 
.14430 
14363 
.14296 
.14229 
.14162 
.14095 
.14028 

90357 
90410 
90463 
90516 
90569 
90621 
90674 
90727 
90781 
90834 

.10672 
.10607 
.10543 
.10478 
.10414 
.10349 
.10285 
.10220 
.10156 
.10091 

93578 
93633 
93688 
93742 
93797 
93852 
93906 
93961 
94016 
94071 

.06862  54 
06800  53 
.06738  52 
.06676  51 
.06613  50 
.06551  49 
.06489  48 
.06427  47 
.06365  46 
.06303  45 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

84706 
84756 
84806 
84856 
84906 
84956 
85006 
85057 
85107 
85157 
85207 
85257 
85303 
85358 
85408 

.18055 
.17986 
.17916 
.17846 
.17777 
.17708 
1.17638 
1.17569 
1.17500 
1.17430 
1.17361 
1.17292 
1.17223 
1.17154 
1.17085 

87749 
87801 
87852 
87904 
87955 
88007 
88059 
88110 
88162 
88214 
88265 
88317 
88369 
88421 
88473 

.13961 
1.13894 
1.13828 
1.13761 
1.13694 
1.13627 
1.13561 
1.13494 
1.13428 
1.13361 
1.13295 
1.13228 
1.13162 
1.13096 
1.13029 

90887 
90940 
90993 
91046 
91099 
91153 
91206 
91259 
91313 
91366 
91419 
91473 
91526 
91580 
91633 

.10027 
.09963 
.09899 
.09834 
1.09770 
1.09706 
1.09642 
1.09578 
1.09514 
1.09450 
1.09386 
1.09322 
1.09258 
1.09195 
1.09131 

94125 
94180 
94235 
94290 
94345 
94400 
94455 
94510 
94565 
94620 
94676 
94731 
94786 
94841 
94896 

.06241  44 
1.06179  43 
1.06117  42 
1.06056  41 
1.05994  40 
1.05932  39 
1.05870  38 
1.05809  37 
1.05747  36 
1.05685  36 
1.05624  34 
1.05562  33 
1.05501  35 
1.05439  31 
1.05378  3f 

31 
32 
33 
34 
35 
36 
37 

P 

4 

43 
44 

4 

85458 
85509 
85559 
85609 
85660 
85710 
.85761 
.85811 
.85862 
85912 
.85963 
.86014 
.86064 
.86115 
.8616 

1.17016 
1.16947 
1.16878 
1.16809 
1.16741 
1.16672 
1.16603 
1.16535 
1.16466 
1.18398 
1.1632 
1.1626 
1.1619 
1.1612 
1.1605 

88524 
88576 
88628 
88680 
.88732 
.88784 
.88836 
.88888 
.88940 
.88992 
.89045 
.89097 
.89149 
.89201 
.89253 

1.12963 
1.12897 
1.12831 
1.12765 
1.12699 
1.12633 
1.12567 
1.1250 
1.12435 
1.1236 
1.1230 
1.12238 
1.1217 
1.12106 
1.1204 

91637 
91740 
.91794 
.91847 
.91901 
.91955 
.92008 
.92062 
.92116 
.92170 
.92224 
.92277 
.92331 
.92385 
.92439 

1.09067 
1.09003 
1.08940 
1.08876 
1.08813 
1.08749 
1.08686 
1.0862 
1.0855 
1.0849 
1.0843 
1.0836 
1.0830 
1.0824 
1.0817 

94952 
95007 
.95062 
.95118 
.95173 
.95229 
.95284 
.95340 
.95395 
.95451 
.95506 
.95562 
.95618 
.95673 
.9572 

1.05317  2£ 
1.05255  2E 
1.05194  2" 
1.05133  2( 
1.05072  21 
1.05010  % 
1.04949  2! 
1.04888  21 
1.04827  2 
1.04766  21 
1.04705 
1.04644 
1.04583 
1.04522 
1.04461 

4 
4 
4 
4 
50 
5 

! 

54 

6 

1 

1 

.8621 
.8626 
.8631 
.86368 
.8641 
.8647 
.8652 
.8657 
.86623 
.8667 
.8672 
.8677 
.8682 
.8687 
.8692 

1.1598 
1.1591 
1.1585 
1.15783 
1.1571 
1.1564 
1,1557 
1.1551 
1.1544 
1.1537 
1.1530S 
1.1524 
1.1517 
1.15104 
1.1503 

.89306 
.89358 
.89410 
.8946 
.8951 
.8956 
.8962 
.8967 
.8972 
.8977 
.8983 
.8988 
.8993 
.8998 
.9004 

1.1197 
1.1190 
1.1184 
1.1177 
1.1171 
1.1164 
1.1158 
1.1151 
1.1145 
1.1138 
1.1132 
1.1125 
1.1119 
1.111 
1.1106 

.92493 
.92547 
.9260 
.9265 
.9270 
.9276 
.9281 
.9287 
.9292 
.9298 
.93034 
.93088 
.9314 
.93197 
.9325 

1.0811 
1.0805 
1.0799 
1.0792 
1.07864 
1.0780 
1.0773 
1.0767 
1.076 
1.0755 
1.0748 
1.0742 
1.073 
1.072 
1.072 

95785 
.9584 
.9589 
9595 
.96008 
.9606 
.9612 
.9617 
.96232 
.9628 
.9634 
.9640 
.9645 
.9651 
.9656 

1.04401 
1.04340 
1.04279 
1.04218 
1.04158  1 
1.04097 
1.04036 
1.03976 
1.03915 
1.03855 
1.03794 
1.03734 
1.03674 
1.03613 
1.03553 

Gotang 

Tang 

Cotau 

Tang 

Cotan 

Tant 

Cotan 

Tang.  1 

490 

48° 

470 

460 

TABLE    XVII.       NATURAL   TANGENTS   AND    COTANGENTS.  30 7 


44° 

44° 

44° 

M. 

Tang. 

Cotang. 

31. 

M. 

Tang. 

Cotang. 

M. 

M. 

Tang. 

Cotang. 

M. 

0 

.96569 

1.03553 

60 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

1.02295 

89 

41 

.98901 

1.01112 

19 

2 

.96681 

1.03433 

58 

22 

.97813 

1.02236 

88 

42 

.98958 

1.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

1.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

86 

44 

.99073 

1.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

1.02057 

86 

45 

.99131 

1.00876 

15 

6 

.96907 

1.03192 

54 

20 

.98041 

1.01998 

34 

40 

.99189 

1.00818 

14 

7 

.96963 

1.03132 

53 

27 

.98098 

1.01939 

83 

47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

93 

28 

.98155 

1.01879 

82 

48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31 

4<J 

.99362 

1.00642 

11 

10 

.97133 

1.02952 

50 

80 

.98270 

1.01761 

80 

50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

31 

.98327 

1.01702 

29 

51 

.99478 

1.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

1.01642 

28 

52 

.99536 

1.00467 

8 

13 

.97302 

1.02772 

47 

83 

.98441 

1.01583 

27 

58 

.99594 

1.00408 

7 

14 

.97359 

1.02713 

4(5 

34 

.98499 

1.01524 

20 

54 

.99652 

1.00350 

6 

15 

.97416 

1.02653 

45 

35 

.9&556 

1.01465 

25 

56 

.99710 

1.00291 

5 

16 

.97472 

1.02593 

44 

86 

.98613 

1.01406 

24 

56 

.99768 

1.00233 

4 

17 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

23 

57 

.99826 

1.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

22 

58 

.99884 

1.00116 

2 

19 

.97643   I  1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700      1.02355 

40 

40 

.98843 

1.01170 

90 

60 

1.00000 

1.00000 

0 

M. 

CotangJ  Tang. 

31. 

31. 

Cotang. 

Tang. 

31. 

31. 

Cotang. 

Tang: 

M. 

45° 

45° 

45° 

TABLE    XVIII.       FRENCH    AND    ENGLISH   MEASURES. 


ES. 


AS 


GH 


W 


>  I 

M  i 

w  «i 

^  w 

W  O 

<1  525 

<q  w 

EH  P* 


E  TABLE 


i 
i 


f  S 


J3    S 

—-« 


^    O 


** 

•-- 


H 


!•--    ai 


11 

H 

I!! 

.00     'i 


^ 

WM       M       M 


f  Ji* 

ll! 

!!! 

«aci^3 
rda):*3 


•  r4  O  |g  r4  O  , 


T-iOO»OOOO5 


(M«OCOCCt--rHCO«O 


§ 

a 

2 

» 


,6 

is 

3 


TABLE    XIX. 


METRIC  CURVE   TABLE. 


TABLE  XIX.   METRIC  CURVE  TABLE. 


Def  .  angle, 
20m. 
chords. 

Radius 
in  metres. 

Ordinates. 

Tangent 
deflection. 

Def.  angle, 
20  m. 
chords. 

m. 

f». 

0  10 

3437.75 

.015 

.011 

.058 

0  10 

20 

1718.88 

.029 

.022 

.116 

20 

30 

1145.93 

.044 

.033 

.175 

30 

40 

859.46 

.058 

.044 

.233 

40 

50 

687.57 

.073 

.055 

.291 

50 

1    0 

572.99 

.087 

.065 

.349 

1    0 

10 

491.14 

.102 

.076 

.407 

10 

20 

429.76 

.116 

.087 

.465 

20 

30 

382.02 

.131 

.098 

.524 

30 

40 

343.82 

.145 

.109 

.582 

40 

50 

312.58 

.160 

.120 

.640 

50 

2    0 

286.54 

.175 

.131 

.698 

2    0 

10 

264.51 

.189 

.142 

.756 

10 

20 

245.62 

.204 

.153 

.814 

20 

30 

229.26 

.218 

.164 

.872 

30 

40 

214.94 

.233 

.175 

.931 

40 

50 

202.30 

.247 

.186 

.989 

50 

3    0 

191.07 

.262 

.196 

1.047 

3    0 

10 

181.03 

.276 

.207 

1.105 

10 

20 

171.98 

.291 

.218 

1.163 

20 

30 

163.80 

.306 

.229 

1.221 

30 

40 

156.37 

.320 

.240 

1.279 

40 

50 

149.58 

.335 

.251 

1.337 

50 

4    0 

143.36 

.349 

.262 

1.395 

4    0 

10 

137.63 

.364 

.273 

1.453 

10 

20 

132.35 

.378 

.284 

1.511 

20 

30 

127.45 

.393 

.295 

1.569 

30 

40 

122.91 

.407 

.306 

1.627 

40 

50 

118.68 

.422 

.317 

1.685 

50 

5    0 

114.74 

.437 

.328 

1.743 

5    0 

20 

107.58 

.466 

.349 

1.859 

20 

40 

101.28 

.495 

.371 

1.975 

40 

6    0 

95.67 

.524 

.393 

2.091 

6    0 

20 

90.65 

.553 

.415 

2.206 

20 

40 

86.14 

.582 

.437 

2.322 

40 

7    0 

82.06 

.612 

.459 

2.437 

7    0 

20 

78.34 

.641 

.481 

2.553 

20 

40 

74.96 

.670 

.503 

2.668 

40 

8    0 

71.85 

.699 

.525 

2.783 

8    0 

20 

69.00 

.729 

.547 

2.899 

20 

40 

66.36 

.758 

.569 

3.014 

40 

9    0 

63.92 

.787 

.591 

3.129 

9    0 

20 

61.66 

.816 

.613 

3.244 

20 

40 

59.55 

.846 

.635 

3.358 

40 

10    0 

57.59 

.875 

.657 

3.473 

10    0 

USE  OF  TABLES  I.,  II,  III,  AND  IV. 
FOE  METEIC  CUEVES. 

THE  metric  curve  table  here  given  corresponds  to  Table  I.,  ex- 
cept that  the  ordinates  for  curving  rails  are  omitted.  The  deflec- 
tion angles,  denoted  by  Z>,  are  for  chords  of  20  metres.  The  radii 

are,  therefore,  computed  by  the  formula  72  =    .         .    In  Table  I 

sin.  D 

the  radii  are  computed  by  the  formula  R  =  -^- .    The  radii  in 

the  metric  table  are,  therefore,  each  one-fifth  or  .2  of  the  radii  in 
Table  I.  for  the  same  deflection  angle.  Moreover,  since  the  ordi- 
nates given  above  and  the  tangent  deflections  vary  only  with  the 
radii,  these  ordinates  and  the  tangent  deflections  may  also  be  ob- 
tained from  Table  I.  by  simply  multiplying  the  corresponding 
quantities  by  .2,  keeping  in  mind  that  corresponding  quantities 
are  those  belonging  to  the  same  deflection  angle.  Table  I.,  ex- 
cept in  regard  to  ordinates  for  rails,  may,  therefore,  be  used  for 
metric  curves  by  simply  multiplying  corresponding  quantities 
by  .2.  The  metre  will,  of  course,  be  the  unit  of  the  resulting 
quantities. 

Example.  Given  in  a  metric  curve  D  =  3°  10',  to  find  R  and 
the  ordinates  ra  and  £  m.  In  Table  I.,  R  =  905.13,  m  =  1.382,  and 
f  m  —  1.037.  Multiplying  these  values  by  .2,  we  have  for  the 
metric  curve  R  =  181.03,  m  —  .276,  f  m  =  .207,  as  in  Table  XIX. 

Since  the  Long  Chords  of  Table  II.  for  the  same  deflection  an- 
gle vary  directly  with  the  radii,  we  may  use  this  table  for  metric 
curves  by  multiplying  the  values  there  found  by  .2.  We  thus  ob- 
tain in  metres  the  length  of  corresponding  long  chords  in  metric 
curves. 

Example.  Given  in  a  metric  curve  D  =  2°  20',  to  find  the  long 
chord  for  five  stations.  From  Table  II.  we  have  for  an  ordinary 
curve  the  long  chord  —  496.689.  Multiplying  by  .2,  we  have  the 
required  long  chord  in  the  metric  curve  =  99.338  metres. 

Tables  III.  and  IV.  may  also  be  used  for  metric  curves,  as  all  the 
quantities  vary  only  with  the  radii.  Therefore,  using  the  same 


312  USE  OF  TABLES. 

deflection  angle,  we  convert  these  tables  into  metric  tables  by 
multiplying  corresponding  quantities  by  .2,  the  ratio  of  the  radii. 
First  find  T  and  b  from  the  tables,  as  for  an  ordinary  curve,  and 
multiply  the  values  so  found  by  .2  to  obtain  T  and  b  for  the  cor- 
responding metric  curve. 

Example.     Given  in  a  metric  curve  2  =  90°  and  D  =  10°,  to 
find  T  and  b.     From  the  tables  we  should  have  for  an  ordinary 


curve    T  =  +  1.45  =  287.935  and  b  =          '    +  .603  = 

119.268.    These  values  multiplied  by  .2  give  for  the  metric  curve 
T—  57.587  metres  and  b  =  23.854  metres. 

It  is  obvious  that  if  chords  of  10  metres  were  used  in  laying 
out  a  metric  curve,  the  multiplier,  as  used  above,  would  be  .1,  and 
that  if  chords  of  30  metres  were  used,  the  multiplier  would  be  .3. 


(46) 


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